I don't mind the idea, but it could cause some balance issues at my table, particularly with the Abjuration wizard and the Rogue.
If I were going to allow multiple reactions, I'd have to find a way to slow it down and let it be introduced gradually. Maybe as a feat?
Lightning Reflexes or Whatever
Prerequisite: Level 4 maybe
Your character has quicker-than-average reflexes, allowing them to react to danger and take action more quickly. You gain the following benefits:
Your Dexterity increases by 1, to a max of 20
You may take one additional Reaction every round.
You may take his feat multiple times, and its effects stack.
Eh...(shrug) I dunno. I think it's a lot more powerful than the OP anticipates, but it's probably not powerful enough to mandate a feat or several feats. It really depends on the players and how hard they push the envelope of game optimization.
And what's good for the goose is good for the gander. If I allowed players to take multiple reactions, I'd allow a handful of high-dex or skirmish-type monsters (like will o'wisps and kobolds) to do it also. Not every monster, but just enough to keep things spicy.