You know that situation. You have a player or players doing something awesome. You give them an inspiration. And they say : I already have one.
It sucks because the system is made that you can only have one at a time. Sure it encourages them to use them, but not always. And if a player is awesome twice without a big die roll he just won't have the time to use it.
One time I was fed up and made this item on the fly : the bottled inspiration.
Its an item that you gain when you get an inspiration and already have one. Its weightless. It takes an action to "drink" it and regain an inspiration, so you cannot do it back to back. And there is no limit to how many you can carry. One of my players made it to 8 one time.
I never had the displeasure of not being able to recompense a player and its been over 2 years using them without drawbacks.
I highly suggest to think of this next time you hear that sad response of already having one.
I think it's a good idea, not dissimilar to WBW's Potion of Advantage. Ultimately it's up to you, as it is every DM, to decide if you want players to be able to stack inspiration in that manner, but as you pointed out, allowing players to stock inspiration tends to discourage them from using it unless they have a stockpile already, because they don't lose anything from not using it.