In my opinion, the best compromise is to let him try one of those temporary tattoos that last a week or two, but with one condition: you choose it together and you think carefully about the design beforehand. It's not an unconditional surrender, it's a "okay, let's see how you really feel about this". A lot of times kids that age mainly want to feel heard when they're curious about trying new things, not so much the tattoo itself.
What doesn't sit right with me about the previous answer is painting a veto as counterproductive. It's not exactly like that: it depends on how you frame it. If you say "absolutely not, stop talking about it" of course that creates frustration. But if you say "not now, maybe when you're older" or "let's try this kind first", that's not a blind ban, it's a boundary. Fifteen-year-olds need clear limits, not parents who give them free rein to avoid conflict.
What really matters is understanding why he wants it: does he want to feel part of a group, is he going through an aesthetic phase, or is he genuinely fascinated by tattoos as a form of self-expression? Once you know the answer you can adjust your approach. And if you decide to allow it, keep it small, nothing outlandish, so if a week later he's over it (spoiler: probably) you won't be thinking "we should've stopped him".