Are there any motherboards out there that will let me install 5-6 GPU cards?

Specifically, I'm looking for GPU cards that take up 2 slots, so spacing is an issue.

Answer

There is the ASUS P6T6 WS Revolution.

6 PCI-E Gen2 x16 IO onboard
True @16 3-Way SLI
True 16+2 Power Phase Design

^{Er, you say cost-effective being the consideration? Cost tracking is a different ball-game with 6 GPU cards on your system...}
This is not a new board -- here is a 3 year old review,

If you're the type of power user that must have 3-Way SLI or even perhaps just have three graphics cards in your system, you don't currently need three full X16 Gen2 PCI Express-enabled slots to get top performance from a tri-GPU setup. As we demonstrated in our high-end gaming benchmarks, you definitely would be short-changing yourself with anything less than a X16 and a pair of X8-capable slots but essentially the added cost of NVIDIA's NF200 chip won't buy you any additional gaming performance, at least currently.

^{And that is March 2009}

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More recently, Asus Rampage III Extreme,

4 x PCIe 2.0 x16 (dual x16 or x16, x8, x8 or quad x8)

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Update from your comments:

1. A dual slot card will take up two slots (by definition), but do look at its capabilities (GeForce GTX 680)


2. Looking at your plans with the hardware, I'd still suggest going for an ready-built server-class machine; and before you give up on the price figures, think about the secondary processing you'll need to do on such a hand-built machine (think heat and power again).