When I started using Ubuntu 10.04 last summer I thought that because it was a LTS release I could simply install it and forget about it for a couple of years, except for running regular security updates.

However, I now find that this is impossible. The main problem is that third party applications aren't updated in the Software Center - the version of the software packaged for 10.04 is the version that remains available, even as new versions are released. The only way to upgrade the application is to use an "untrusted PPA" from Launchpad, or to compile from source, taking care to also update any new dependencies.

Is there a Linux distro that makes this easier? It would be ideal if the official installer, the equivalent of Ubuntu's Software Center, would offer newer versions of third party applications, without recourse to PPAs. Does such a distro exist?

https://help.ubuntu.com/community/UpgradeNotes has instructions on upgrading from 10.04 LTS to 10.10 which is, I assume, what you are after. You can also follow the instructions at http://www.ubuntu.com/desktop/get-ubuntu/upgrade (I just did!)

Upgrading the system is what I'd like to avoid, just upgrade the applications, like Gimp, etc. I don't like using "untrusted PPAs" and I've run into issues when compiling... What would solve it for me is an easy way of installing newer versions of software from official repositories without upgrading the system. That's what I (erroneously) thought a Long Term Support release meant. So now I'm curious if another distro handles this better.

For all distros you can upgrade (or not) as you wish. Have a look at apt-pinning, you can force the system to keep a particular piece of software at a particular version and it will then be skipped. However, by doing that you run the risk of not getting security updates for that piece of software. All LTS means is that your system will be stable, which often equates to using non-current software as that software won't be deemed stable until it's been out in the wild for a while!