I've heard this argument so many times. It's cliche and annoying, but more importantly its wrong. A teenage kid mentions to an adult that he's thinking about getting a tattoo. Then the adult points out how different his body shape is since he was young. "If I had gotten a tattoo when I was your age, just imagine what a mess it would look like now." He might even have a picture of himself as a young lad with a thin profile, and muscled arms. Now he has a beer belly, a wrinkled face, and flabby arms. The implication here is that if he had gotten a tattoo when he was young, by now it would be so distorted that you'd barely recognize it. The young kid should think twice before getting a tattoo. Don't do today what you'll regret tomorrow. Is this a good reason to not get a tattoo?
Nope and here's why. First, this argument ignores what economists call a relative cost change or an Alchian and Allen effect. If the kid admits that his body shape will change for the worse then this is a case for getting a tattoo not for abstaining from one. If your going to get old, fat, flabby, and wrinkled anyways then being old, fat, flabby, wrinkled, and having a messed up tattoo is not a very far jump.
If getting old with a tattoo makes you marginally ugly but getting old makes you ugly anyways, then from the perspective of the person making the decision today, it's cheaper to get the tattoo once you admit to yourself that you're going to be old and ugly. In simpler terms, if you're eighty years old and you think that your tattoo will look silly when you take your shirt off, chances are you're gonna look pretty silly with your shirt off anyways, so why not have fun while you can?
Second, if there are actions you can take to control the changes in your body type, then getting permanent skin art could be an attempt to credibly commit to those types of behaviors. Over eating, not working out, and overexposure to the sun are all marginally more expensive with a tattoo than without. So getting more tattoos could influence the way you choose those behaviors and maybe even effect your future body shape.
Now I'm not saying that every teenager should run out and get tattoos. All I'm saying is that this old cliche is messed up and probably encourages kids to get tattoos more often than it discourages them.