So should you ask for a show quality puppy? If you're planning to show the dog, then obviously YES!!! But be prepared to convince the breeder that you are serious about showing. The breeder may want to co-own the puppy, to ensure that you stay in contact so the breeder can mentor you.
if you're just looking for a beautiful, healthy pet then it doesn't have to be show quality!
Real show breeders want ALL their puppies to go to wonderful, loving homes - but for the best puppies they also usually prefer that they go to people who intend to show them. This isn't just for ego-boosting or vanity although there may be some of that. But reputable breeders spend a lot of time and energy watching and evaluating other dogs entered in shows. To see what crosses seem to work, and what don't work. To see what certain family lines are like. It's all part of the education process, and also part of the complex process of selecting the best animals to continue the breed. So of course they want to try to be sure that their best dogs get "out there" to be seen, and evaluated in thier turn. That is only reasonable. And it's why a reputable breeder may refuse to sell you the "pick of the litter" when all you want is a nice pet.
But sometimes in a nice litter there will be many truly show quality puppies, and since not very many people want to show their dogs you can still wind up with a show quality dog even though he'll never set foot in a show ring.
WARNING: there are many unscrupulous breeders out there who are happy to claim that a puppy is "show quality" just so they can charge top dollar for it. That's why you have to be careful about asking for a show quality puppy. If a breeder is advertising "show quality" puppies, but that breeder doesn't show - and there are no Champions in the first generation or two of the pedigree (parents or grandparents) then it's a scam.
But maybe you're afraid that a "pet" quality - i.e. a non-show-quality - puppy will be inferior in some important way. It's important to remember that even a "pet" quality puppy from a well-bred litter will still be a beautiful dog - in fact, you may not be able to tell the difference between your "pet" and his "show quality" sibling. The differences are often very subtle and require an expert eye to see them. And reputable breeders put just as much time, effort and love into socializing and training and raising their "pet" quality puppies as their "show" quality pups.
Better yet, consider adopting a wonderful Great Dane from a Great Dane Rescue. Rescues occasionally get young puppies, but more often get young adult dogs who had the misfortune to belong to people who didn't think ahead, and weren't prepared for life with a Giant breed. These dogs are usually housebroken and often past the chewing stage, so can be easier to live with than a young puppy. They just want someone to love!
You can do a search for a Dane Rescue in your area, or visit the GDCA website for a listing.