Let other people brag, then pass judgment on their bragging.
GUY: I blah blah blah (list of his big achievements).
YOU: That's really impressive, dude. You've done a lot. I admire that.
Be sincere about it. Don't do it in a tooling way. Actually mean it.
The guy will like you, because you are smart enough to realize how awesome he is. At the same time, he will be thirsty for more of your approval.
He will not totally understand why, because he is the one with all the achievements, and he may not even know much about you, but suddenly he will find himself seeking your approval more and more. Everyone else will regard you as the higher status guy too.
I have known guys who were so good at this that they had basically no (ordinary) achievements of their own, but they were surrounded with various high value, high achieving people who were constantly deferring to them for approval, meanwhile the alpha guy is just a really good, confident, dominant guy, who cheerfully praises and gives out esteem to other people for their various accomplishments, meanwhile declining any praise people try to throw his way with humbleness.
I knew a kid like this in high school... he wasn't top of the class, wasn't an athlete, wasn't anything remarkable in an ordinary sense, but he was THE most popular kid in school. All the cool kids revolved around him. He wasn't especially good-looking, I don't think (he had a big nose). His voice was kind of high and he didn't work out (skinny). But he was a REALLY good guy, decent sense of humor, and he was so positive and just really loved people and was so good at recognizing exactly what you were good at and praising you for it. I was this aloof kid who everybody treated as this outsider weird/cool guy... well, that other kid was the only one in high school that made me feel like I really wanted his approval, and he did it by just every so often commenting on something I did and telling me how cool that was or how much he admired that I did X or how impressive it was I was able to Y or how he noticed I was really good at Z.
I had a buddy like this in college. Constantly around high status people, rich people, successful people, good-looking girls, and he is very, very good at recognizing people's strengths and, maybe more importantly, what they WANT people to view as their strengths, and offering them sincere praise on those. He also recognizes people's weaknesses but he keeps that to himself unless you're close with him. The dude cuts through bullshit like a hot knife through butter.
I had a girlfriend like this too.
I always have to remind myself to be less stingy with praise. It's such a mighty social super power. If you can train yourself to comment on things people are good at and give them sincere praise for it, you can get a lot of people liking you very quickly, and you don't even really have to do much of anything -- other than be a guy who allows other people to feel recognized for what they want to feel recognized for.
Aside from that, it simply positions you at the top of the hierarchy. As a wise man once told me, "Alphas judge." i.e., the guy who is the alpha male of the group passes judgment on things and approves of them (or not).
Thus why people chase approval from the guy who shows them he is capable of passing out approval on things.
(obviously, must be calibrated... approve/pass judgment more low-key when you are lower status. Once you're high status you can really build a guy up by talking him up, really highlighting his good points, etc.)
Chase