The absorb elements spell says

The spell captures some of the incoming energy, lessening its effect
on you and storing it for your next melee attack. You have resistance
to the triggering damage type until the start of your next turn. Also,
the first time you hit with a melee attack on your next turn, the
target takes an extra 1d6 damage of the triggering type, and the spell
ends.

At Higher Levels. When you cast this spell using a spell slot of 2nd
level or higher, the extra damage increases by 1d6 for each slot level
above 1st.

So if my Sorcerer is hit with a fire bolt from the enemy mage, I use my reaction to cast absorb elements at 3rd level so I now have 3d6 fire damage stored for my next melee attack, I then use my action to cast a 1st-level spell and my turn ends, resetting my reaction. Another enemy hits me with chain lightning and I use absorb elements again at 3rd level.

Do I now have 3d6 fire + 3d6 lightning or just 3d6 lightning stored for my next melee attack?

No, the extra damage is only added to one attack on your next turn.

Read the Absorb Elements text carefully:

Also, the first time you hit with a melee attack on your next turn, the target takes an extra 1d6 damage of the triggering type, and the spell ends.

(emphasis added by me)

The spell only provides a benefit during the turn after you cast Absorb Elements. You cannot "store" the extra damage across multiple turns.

If the Sorcerer did not make a melee attack roll on their next turn, then the benefit of Absorb Elements is lost at the end of that turn.

Note: It is possible to cast Absorb Elements twice such that the durations overlap, but the damage won't stack. This is because a character's Reaction resets at the start of their turn, not at the end:

When you take a Reaction, you can’t take another one until the start of your next turn.

For example, suppose the Sorcerer takes fire damage before their turn, using their reaction to cast Absorb Elements. Then on the Sorcerer's turn, once their reaction resets, they somehow take fire damage again, and spend their reaction (again) to cast Absorb Elements (again). This would mean having multiple castings of the same spell, but their benefits do not stack:

The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect--such as the highest bonus--from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.

If the Sorcerer proceeds to make an attack on this same turn, then only the "most recent" or "most potent" casting of Absorb Elements would apply to that attack.

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.