I have a first generation iPod touch (SW version 1.1.5). I believe that this version does not allow me to put Apps on the touch. Is it possible to purchase an upgrade? If so, would I then be able to use apps?
Yes. You can install iOS 3.1.3, which is the highest version for the 1G iPod Touch.
Purchasing iOS 3.1 Software Update for iPod touch (1st generation)
is there any way that you can get it without the purchase?
Not legally. Sorry. It is only a couple of dollars. ;)
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