The situation you described isn't possible.
You must use your reaction to cast absorb elements, and its effect benefits your first hit with a melee attack on your next turn, regardless of which turn your reaction occurs on (even your own).
You regain your reaction at the start of your turn, not the end of your turn (see the rules on Reactions):
When you take a reaction, you can't take another one until the start of your next turn.
If you cast absorb elements using your reaction in response to taking fire damage, then you could benefit from the added fire damage on the first hit on your next turn. Since your reaction is gone now, you will be unable to cast absorb elements again before your next turn starts. You regain your reaction at the start of your next turn, but even if you manage to use your reaction on that same turn to cast absorb elements again, the second casting would benefit your first hit on the next turn after that one, not on the current turn.
To put it briefly: the start of any given "next turn" can be preceded by at most one casting of absorb elements, since you have at most one reaction prior to that "next turn."
So, no, it's not possible for the added damage effect of two absorb elements castings to stack on a single hit because the two castings will be forced to benefit hits made on successive turns, not the same turn.
There aren't any edge cases to the above answer that I'm aware of.
You only have one reaction to use, barring special rules that would violate the game's design principles, according to the Dungeon Master's Guide section on Creating New Character Options (paraphrased for brevity):
Beware of adding anything to your game that allows a character to use more than one reaction per round. Rules and game elements that override the rules for reactions can seriously unbalance or overcomplicate your game.
There aren't any magic weapons, epic boons, or other special features I'm aware of that allow you to cast a reaction spell using something other than a reaction in the official rules (as of the current errata accounting for spell scroll casting times).
This is why my answer doesn't address what would happen if one hit were to benefit twice from absorb elements: it's a hypothetical that falls outside the rules.