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Wiktionary
braggy

a. Prone to brag

WordNet
braggy

adj. exhibiting self-importance; "big talk" [syn: boastful, braggart(a), bragging(a), big, cock-a-hoop, crowing, self-aggrandizing, self-aggrandising]

Usage examples of "braggy".

So it was that evening, for after introducing himself with a brief, braggy speech, Vice Principal Nero stood on the stage of the auditorium and began playing his sonata for the first time.

The dog yapped at them, bright-eyed and happy in a braggy sort of way.

Not to sound braggy, but I am quite beloved by the American people for saving the life of their leader.