I want a new computer chair.

Can anyone recommend a place with a big selection of good chairs? The cookie cutter places like Staples, office depot, etc. have a lot of chairs but mostly just cheap junk.

The chairs you'll find at the Office Depot/Staples/Office Max retail level, at least the ones they have available in store and on display, will be lower end.  However, take a look at their business catalogs, which is where the good stuff is hiding.  All of the office supply stores have a "business account" side, which is completely separate from the retail side. Usually the store associate will be able to reach out to the business catalog side of the company and get one ordered for you.

I just would hate to order a chair without sitting in it first.

If youre near Portland, try City Liquidators. Lots of different choices and reasonable prices.