This is a follow-up to the question on the Ramanujan constant.

Can one find an asymptotic family of shockingly near-integers obtained by raising e to some simple algebraic number, in the spirit of Ramanujan's $|\exp(\pi \sqrt{163}) - \text{ some integer }| \le 10^{-12}$? Single numerical coincidences don't impress me (and for good reason too ...).

Is there a sequence of such near-integers, whose nearness to integers is closer than what you know exists by pigeonholing exp(alpha) for all alpha of bounded "height" or "complexity"?