Any of the big mountains in washington (Olympus, Baker, Ranier) have glaciers that provide conditions ripe for mountaineering gear. That said, I don't know as there is a more beautiful spot on earth than overlooking the glaciers on Olympus (might be a little biased. I've hiked several times up the Hoh river valley to Glacier meadows and camped and spent several days scrambling around Mt Olympus. Played on the glaciers some, brought an ice axe and rope, but missed ascending it when I was in boy scouts years ago by 2 years (needed to be 14 to make the trip to the top and i was 12, so I hiked in and wandered around the snow fields and surrounding peaks.)

I've always had good luck in the Hoh river valley, it's always rained on me at least once, but in july/august it's always been warm enough that I hike in shorts and a t-shirt and then change to dry clothes when I get to camp.

If you are looking to summit something that doesn't require gear, there are several peaks that don't have glaciers on them around mt olympus. Without my maps I couldn't name any other than goat peak. I remember that one because the ranger last time I was up there told us a story about it. It's a nice area, I don't think you'd be dissapointed hiking up that way and seeing the mountain up close. Unless you are in stellar shape it's 2 days to Glacier meadows, 17 miles but 2 days. Last time I went we hiked 5 miles the first evening (left from Seattle late morning) and camped at 5 mile island then took the next 12 miles the following day. I thought we were going to have to make camp for the night, my hiking buddy was having a rough time by the time we got it. It's fairly steep for the last 8 miles in.

Let me know if you want more info.