Here are four ideas that are potentially going to be refused:
1. White Ink Tattoo
I admit, they look pretty cool. However, white ink tattoos only look good on very, very fair skin, so us tanner people won't get quite the same effect. Also, you never know how it's going to heal; it could end up looking yellowish, or like an ornate pink scar. For those who really want a white ink tattoo: choose your artist carefully. I know I say that a lot, but it's very easy to fuck up a white ink tattoo! If your tattoo artist begins using a blue-ink stencil -stop them. They don't know what they're doing. The blue will mix in with the white and you'll have a blobby mess.
2. Special parts.
I won't post a picture for this one, because you know what they are. Before you get all upset when no one in the tattoo shop wants to tattoo your "mommy and daddy button," consider this: Would you honestly want someone's (possibly stinky) junk up close, in your face, for a couple hours? If you find someone willing to put up with tattooing you "down there," keep in mind it's probably going to cost you wayyyy more than any other place you've been tattooed (and I don't blame any tattoo artist for hustling you).
If this photo sparks your interest you can read her story here
3. Face Tattoos.
In this situation, think of that tattoo artist as your conscience. They recognize that customers can be impulsive and naive, so they may refuse tattoos that are on the face. An artist may go through a long process of interviewing the client to make sure that it's what they really want, and asking what the client does for a living. Face tattoos are very noticeable, they can't be hidden, and tattoo artists who make a big deal out of a doing a face tattoo are only trying to protect the client.
4. Inner Lip Tattoos.
This is one I've wanted myself. Yes, I will go on record and admit it. But I don't have it yet... and that's because inner lip tattoos are very difficult to do. They have to be free-handed. It's also very difficult for the skin there to "hold" the ink. In fact, many inner lip tattoos "disappear" within a year; they require periodical touch-ups.