You can rent a canoe for cheap at UW (like $7 an hour I think) and paddle across to the botanical gardens - great way to spend an afternoon! Turtles, birds, plants... bring a picnic lunch and haul out for a few minutes.
If you're able to drive/travel from Seattle, the San Juan Islands are neat, or if you don't live near a forested area, you should check out some old growth forests, the trees are impressive. (Sorry, don't have a good old-growth forest to recommend near Seattle, I only know good ones near Bellingham - but others may have ideas.)