Is there a way to get back apps you downloaded but didn't back up?

Hi try this: go to the app store on your iphone and log in using your apple id. Once you have done that, go to where it says update. When you clicked on update it will say purchased. Click on purchased and you are then given two options "on this phone and not on this phone" click on the second one and the apps that you have paid for will appear. I deleted an app on my computer but i was able to get it back this way. If the apps you paid for dont appear using the second option then try the first option.

Note: this wont work if the apps where aqquired using jailbreak.

You should be able to re-download them using iCloud purchases in iTunes.

if you have receipt just contact itunes and ask