Почти has nothing to do with grammatical case. It is not a preposition, it's an adverb.

около десяти = about ten. около is a prep.
с десять = about ten. с is a prep.
приблизительно десяти = approx. ten. it's a prep.

When you find a noun in some other case following почти it is because the case of that noun is governed by some earlier word in the sentence, not by почти.

к примеру-- после почти десяти лет . Десяти is in gen. because of the preposition после, not because of почти.