This will give you what you want. A friend of mine bought last year's model for his cottage. Much better sound than Bose radio. Much less expensive. It has a CD player, tuner and iPod dock built in.
Or, for an equally compact system with better sound quality, for a little more cost but still in your price range, you could get the "integrated receiver" only from that Yamaha mini-system (not the speakers) - the "integrated receiver" is sold separately as Yamaha model CRX-332: http://usa.yamaha.com/products/audio..._b/?mode=model
Leaves you $300-400 for a pair of bookshelf speakers. Denon has the SC-N7 matching speaker. They also have a bookshelf system in the $400 range comparable to the Yamaha. I'm not sure if you can find these in store, but they are available on Amazon.
I was looking at the Bose wave but it is too pricey for my budget right now.
Is there any CD players that are very similar to the Bose ?
I want something that has a simple set up and a really smooth bass at a reasonable price.
I'd look at the various HTIB (Home Theatre In a Box) products, and analyze them as a player of audio CDs. That they will play audio CDs and other media is a slam dunk. I note that portable digital player interfaces are a common feature.
Google is your friend, and going to Amazon and searching on HTIB will get you a lot of alternatives.