What kind of shirt are you looking for --- a button-up type with pockets, or a pullover stretch type?

For the former, just about any all-synthetic shirt is good for me; REI sells them, though you have to look at the tag. Just because it's at REI doesn't mean it's not at least partly cotton (!). A thrift store can sometimes carry such too, but again, look at the tag. I think in recent years there's at least some cotton blended into most dress shirts for comfort (I miss they tacky days of all synthetic for this reason :-)).

For pullover shirts, I've used merino wool, but unless you're concerned about odor on the trail, I prefer capilene. In particular I like the lightest capilene layer, cap 1. I can wear this in quite warm weather, but it's excellent as the next-to-skin layer too. I'm happier with a heavier weight base layer if not too active, but on days when I'll be exercising outdoors a lot, cap 1 is a wonderful base.

I think in general I tend to wear an all-synthetic button-up shirt next to my skin as I like to have pockets. This isn't an issue on days when I expect to be wearing enough layers over the shirt that the pockets are harder to access, however.