I purchased a tv show. I like it so much I was willing give as a gift. Unfortunately the gift button doesn't appear. It seems because I have already purchased it for myself, I am unable to give it as a gift.

It appears to be a 'feature' of the current version of iTunes. If you are doing it on your computer's iTunes then try logging out of your account and you should then get the gift option - you will then be prompted to sign back in as part of the gifting process.

If I go into the Purchased tab on my phone and select the series from the list of my purchases then I get the same as you are seeing i.e. without the Gift option. But if I use the Search tab and search for the series (i.e. find the series in the main part of the store, not from the list of my purchases) then I get the Gift option e.g. via the search screen on my iPhone 4S with iOS 6.1.3 :

If your phone is on iOS 6.1.4 then it's possible that that is causing the gift option not to appear via the Search screen (as my phone is a 4S I can't update to 6.1.4 so I can't check). As a work-around, if you log out of your account by tapping on your id in Settings > iTunes & App Stores and then search for that series, do you then get the 'gift' option - you should then be prompted to sign back in if you tap on the Gift icon.

Signing out, making my selection and then signing back in (when prompted) did the trick.