Installs fine on XP but I want to install it to WinME because that PC has a CRT monitor connected to it instead of a widescreen LCD like my XP/Vista PC has. Only probelm is that the gog.com installer causes a runtime error on WinME. Anyone know a fix for that?

There is no fix. The installer is only compatible with XP and Vista, not with ME, 98, 95 or anything older.

Considering one of the main purposes of this site is to take classic games that do not install/work on modern operating systems and update them so that they do, the fact that they can't run on ME should not seem weird at all. The installer itself is specifically designed to work with XP and Vista because one of the major issues with older games is not so much that they cannot run on a modern OS, but rather that the installer they used to come with doesn't work.

I can install any game on XP, I came here to buy 'good old games' and not necessarily games that the installer requires XP or Vista. Old games look better on my old CRT so it is a bit of an issue for me.

So why not connect the CRT to your XP or Vista machine?
If you came here to buy old games expecting them to run on an old OS, then you should have read a bit more about the site and your purchases first. The purpose of the site is not to simply sell old games in their original form that won't actually work for most of GOG's potential customers, they sell old games that have modified so that they do work for most of the potential customer base. That modification may mean that the game no longer works (or at least won't install) on the OS version it was originally released under, hence why every game sold here lists XP or Vista as a minimum requirement on the game's information page.

Has anything ever been compatible with WinME?

My axe!

Doesn't matter anymore as I dumped WinME and installed Win2KPro to it so will just use Vista or XP in future for GoG games. And WinME wasn't crap if you knew how to stroke it to get some magic out of it. ;)
Oh yea, and one game I got from here made my Vista games folder continually go into search mode when ever that folder was opened. Can't remember which game it was now as I have bought a few from here but it was a bitch to get rid of.
Also, I already have 2 widescreen Samsung LCDs connected to my XP and Vista PCs so don't want to put a clunky CRT on the desk too and ruin the Feng shui.