There's no good reason to get four 290X cards. You can get good scaling up to three GPU's, but very little beyond that.
The short answer is tha you will not find any air cooler that can adequately cool three GPU's side by side with no empty space between them.
A custom water cooling loop would produce the best results, but you'd have to spend about as much on that as the cards themselves to do it properly.
Closed-loop coolers would be a viable option, if you have enough places to mount the radiators.
As for PCIe expanders, I have no experience with them, but I expect you'd spend less money getting a pair of already-water-cooled 295x2 cards.