Where does Itunes keep bought apps?

"Actually, you probably do need to pay for it twice. When you 'bought' it, you (presumably) payed for a single licence for a single device.

Regardless, on OS X Leopard, it should be in:

Library/iTunes

Windows XP:

C:\Documents and Settings\<Your Windows username>\Application Data\
Apple Computer\iTunes\iPhone Software Updates

And in Vista/7:

C:\Users\<Your Windows username>\AppData\Local\Apple Computer\iTunes\
iPhone Software Updates

On the other hand, to see where the iPod touch and iPhone applications are, just right click one and select 'Show in Explorer/Finder'"

My guess is that it is stored in a temporary folder somewhere. I know it sucks, but it might be best just to pay the extra $10 to get it upgraded properly.