Is there a PPA for gvim > 7.4.338?

There is the pi-rho/dev PPA, which has up-to-date vim packages.

(The current default version for trusty is 8.0.0134.)

The PPA is available for currently supported releases.

Vim 8.0 (just released, 2016-09-12) can be installed from a PPA in Ubuntu 14.04 and 16.04.

sudo add-apt-repository ppa:jonathonf/vim
sudo apt-get update
sudo apt-get install vim

Today I found this one

ppa:pkg-vim/vim-daily

It uses the new dailybuild function from launchpad to automatically grab the upstream source and build if there has been a change. So nearly no maintenance needed from ubuntu side.