Technically there is an infinite number of achievements, from the achievements that are awarded every time your rating increases by 100 points, i.e. achievements for bullet 1500, 1600, 1700, and so on. However, there is obviously a limit to how high you can push a rating on chess.com, so you can try to obtain all the achievements as well as the the rating achievements up to about 2900 rapid, 3200 blitz, and 3400 bullet.
Achievement hunters
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I'm interested if there is more people like me, that are hunting down all the achievements (if possible) in this awesome game of chess. Let me know down in the comment, or maybe we could become friends that will hunt together