Although any canine--when raised and handled properly--can be a good companion, the advantage to selecting a dog of a specific breed or type is that you can make some generalizations about how members of that breed typically act, or what their personality is like. Some breeds are typically considered "easier", as a general rule, whereas other are "higher maintenance". Of course, individuals within the breed can vary greatly...but the generally accepted breed traits will give you an idea of what to expect. They offer a glimpse of what the average owner of the average member of that breed is likely to experience, when raising such a dog.

What constitutes a "difficult" dog, depends primarily on what YOU are good at, what your home environment is like, and what behaviours you dislike or don't know how to work with. Some may find an independent dog to be difficult; others may have more trouble with an overly clingy or anxious dog.
A good example would be the pitbull terrier I am babysitting for a few months. He *loves* people, and is obedient and eager to please. He also is very clingy, and does not get along well with other dogs. This makes him a rather troublesome dog for me--I am much happier with my wolf/dog mutts, who are substantially more independent, quieter, get along great with each other...but can be rather destructive, and very timid around people they don't know. However, many dog owners would feel quite the opposite!
I may not like an extremely high energy dog; his vibes may make me anxious or frustrated...yet someone who loves to go running and play frisbee might find the high-energy dog a perfect fit.
No one breed is a good match for everyone, and fanciers of most breeds are quick to tell you that. the dog "difficult"...or is he just difficult for YOU? ;-)

That said, a number of dog breeds/types are generally said to be "high maintenance" in general; these are dogs that would be considered 'advanced level' canines, due to their tendency to possess traits that could be challenging for less-experienced owners to work with. Successful ownership of these dogs usually requires a fair amount of dog experience, good handling skills, the ability to understand canine body language and communication, an above-average level of commitment, and the willingness to be a responsible and conscientious caretaker. All of these things are important when raising ANY dog, but slacking off as the owner of a toy poodle or Golden retriever is less likely to cause problems than failing with a high maintenance dog.

High maintenance dogs tend to have traits that fall into one of several categories:

Strong-willed dogs. These dogs know what they want, and are willing to argue with you (or ignore you) to get it! Breeds whose members often fall into this category are Jack Russell terriers , American pitbull terriers, Alaskan malamutes, chows, Japanese Akitas, working line German shepherds (or Malinois, or the like), some wolfdogs (AKA "wolf hybrids"), some rottweilers, Australian cattle dogs, Catahoula leopard dogs, and other dogs that were bred to work independently. These dogs need a strong, firm, fair leader and a good No Free Lunch/Nothing in Life is Free program. They need an owner who is willing to set and enforce limits. Otherwise, they may end up being dominant over the human members of the household, which opens up a whole world of trouble.

Independent dogs. Many people expect dogs to seem "eager to please", and to have a strong focus on their human's actions and desires. This is not always the case, with dogs bred to work independently of humans and to think for themselves. An independent dog's world does not necessarily revolve around their humans. They have their own agenda, are often less 'needy' towards their people than the average dog, and do not solicit as much attention. They may not work for praise or petting, requiring instead that you reward them with a "salary" of treats or other things of value to them. Clicker training usually works extremely well, for the dog who asks "but, what's in it for ME?" Breeds known for this attribute are Northern breeds such as the Alaskan malamute, Siberian husky, and Samoyed; primitive breeds such as the wolfdog, Basenji, Canaan dog, and New Guinea Singing dog; sighthounds like the Saluki and Afghan hound; and Asian breeds such as the Akita, chow, and Korean jindo.

Timid and/or over-reactive dogs. While fearfulness is largely based on early experiences and poor lines within the breed, some breeds tend to have a larger proportion of shy or skittish dogs. Shy dogs require a trustworthy human leader, a calm and secure environment, above-average early socialization, structured training, and confidence-building exercises. They are often a poor choice for homes with small children, and may show higher levels of reactivity than a more laid-back dog. When improperly handled, shy dogs can become "fear biters" or be difficult to manage in public. Breeds with a higher percentage of timid or reactive dogs include wolfdogs, German shepherds, Border collies, and a number of small dogs such as Chihuahuas. (Side note: wolfdogs are not an actual breed, but rather a 'type'—in the same way that Alaskan huskies or pitbull terriers are a 'type'. Dog types tend to have more variation from animal to animal than that within an established, registered breed.)

Destructive and/or high energy dogs. Labrador retrievers are not usually considered "high maintenance" dogs, but if the worst thing you can think of is having your house or belongings chewed up, then perhaps a lab is not for you! ;-) Any dog can be destructive, especially as a pup...but labs, pitbull terriers, boxers, huskies, malamutes, wolfdogs, German shepherds, and Border collies tend to be notorious chewers. Many of these are also prone to separation anxiety, which can result in massive destruction from an uncrated and unsupervised dog. They also tend to be high energy breeds, which can make it harder to keep their attention during obedience training and can result in all sorts of problem behaviours due to displaced energy and frustration. Many high energy dogs were originally developed to be WORKING dogs, and they need not only a large amount of exercise, but also a job to do (!) and a fair amount of mental stimulation as well. ~As a general rule, working dogs do not make "good pets" in the average home.~

Not necessarily good with children. While children should NEVER be left unsupervised with any dog, there are certain breeds that are a particularly poor choice for a child's pet. Strong-willed dogs are often inclined to become dominant over children, which can lead to "corrective" bites. Timid, under-socialized, or reactive dogs may become frightened by children's actions or rough handling, which can lead to fear-based bites. Dogs with heightened prey drive may chase children who are running and squealing or riding on bicycles, and there have been cases of children killed by such dogs! High-energy dogs may simply knock small children down, scratch them, step on them, or break skin with rough play. It cannot be stressed enough that ANY dog must be supervised with children—-but if you are looking for a companion for your child, it is best to stay away from wolfdogs, chows, Border collies, Rhodesian Ridgebacks , Jack Russell (and similiar) terriers , and Chihuahuas (to name a few) altogether. If buying a puppy, ALWAYS research the breed's typical temperament carefully before settling on a child's pet.

Dog-aggressive or other-animal aggressive. Early socialization makes a tremendous difference in a dog's ability to develop the social skills to get along well with other dogs. Even so, certain breeds of dog are more inclined to fight with other dogs upon maturity—especially others of the same sex. Pitbull terriers, Alaskan malamutes, Akitas and Shiba Inus, Korean Jindos, New Guinea Singing dogs, Jack Russell terriers, Canaan dogs and Basenjis are breeds noted for same-sex aggression. Wolfdogs are notorious for female-on-female fights, although the males tend to be more accepting. Leaving any of these breeds intact greatly increases the likelihood of aggression.

Primitive breeds and terriers also tend towards having an intact prey drive—they have the instinct to not only chase smaller animals, but to follow through with a bite or kill. Unless well-socialized to small animals as a pup, these breeds—along with hunting and working dogs such as beagles, shepherds, Rhodesian ridgebacks, etc--have a higher propensity to chase and 'break' other pets such as rabbits, birds, and cats.

(Please note that that dog-aggression, prey drive, and human-aggression are three separate issues, with three separate underlying causes. A dog can be highly driven to take down chickens, but be wonderful with people. Likewise, a dog can attack and kill other dogs, yet be completely safe with toddlers. However, a human can still be injured when *breaking up a dogfight*, or when retrieving a slain chicken from a food-aggressive dog!)

HIGH levels of owner responsibility and diligence are required, to ensure that a dog who is willing to harm other animals (or humans!), never gets the opportunity to do so. This includes keeping the dog on-leash or in secure containment; checking leash & collar hardware regularly; never allowing small children or others who cannot physically control the dog to walk him; paying attention to your surroundings when out in public; putting a lock on the gate to his yard if the kids cannot be trusted not to leave his gate open, and so on. If the above precautions sound like 'too much trouble' rather than 'solid common sense', then perhaps a high maintenance dog is not for you.

Escape artists. Some dogs are content to stay in the yard by their owner's side, with no fencing at all; for others, a 4 foot picket fence will suffice. Then there are the containment-challengers: dogs who will dig under, jump, climb over, or tear through the average fence in minutes! Escape artists seem to vary as much by individual as by breed, in my experience, but certain breeds have a reputation for being tough to contain. In today's world, containing your dog is no longer optional (!) and the trouble and dangers he can get into while running free are extensive. Children can be chased, other dogs and cats can be attacked, strangers walking through "his" assumed territory can be bitten...or he could simply be hit by a car, or drink antifreeze in the neighbor's garage. Proper containment is an important part of ANY dog owner's responsibility, and if you are not willing or able to put up a strong and sizeable fence, it is advisable to steer away from Northern breed dogs (such as huskies, malamutes, or wolfdogs), pitbull terriers (whose strength and determination are legendary), super-smart dogs like Border collies, and even some smaller dogs such as Basenjis and Jack Russells (who were born to dig)!

The moral of the story is that it is very important to understand your dog's potential issues, based on his heritage...and become proficient at accommodating his needs and working around his weaknesses.
Breed tendencies are just that: tendencies. For instance, I have one wolfdog who is completely reliable offlead, another lazy wolfdog who serves as a pillow for our cats, and a very easygoing male wolfdog whose father was a pure wolf, who can go anywhere, do anything, and was easier to raise than most German shepherds. However, this may not be the experience of the "average" person with the "average" wolfdog.
I have met chows that you could walk up to, and give a big hug and nuzzle them and say "awww, aren't you just a big cute teddy bear!"...BUT, I would no sooner walk up to the average chow on the street and do this, than stick my arm down the garbage disposal! ;-)
I have known of mature pitbull terriers who snuggle and play delightfully with other pitties...but it is not the default behaviour experienced by the majority of pitbull owners. There are exceptions to every rule...but, DO be aware that there are basic predispositions within breeds that are the origin of the anecdotal 'rule'.
With sufficient knowledge and proper handling, you can overcome almost anything a pup's genetics toss out at you...and virtually ALL healthy pups can be raised into safe and rewarding companions. (Some ideas on raising safe dogs HERE.) But, if you have the choice, why acquire a dog who is likely to come with behavioural leanings that are quite different from what you desire in a companion?

Of course, there are many other breeds besides the ones discussed above, that have their difficult moments! This essay is merely intended to get you thinking about YOUR potential breed of choice--how good of a match he will be for you, in what ways he might present a challenge, WHY those behavioural speedbumps occur, and how you can avoid them. You can reduce your odds of failing your dog by choosing a breed that suits your lifestyle, personality, and abilities--be it a well bred pup, or a temperament-tested rescue. High maintenance dogs can be extremely rewarding, but they belong only in the hands of those owners who are willing and able to go the extra mile to do right by them, and turn them into respectable representatives of their breed.