Sure, the original Starbucks is located at Pike Place Market. But I'm not a supporter. There are plenty of other places to get great coffee while at Pike Place. For that matter, there are plenty of places to get better coffee anywhere, but don't get me started.
Local Color is such a place. While waiting for my car to have an oil change, I watched the mob go by from a table at Local Color, drank an excellent latte, listened to jazz, and admired local art and local artisan wares that fill the place. As you can see form the cup, they use Cafe Vita coffee. My latte was smooth and creamy with good foam and tasted toasty.