THIS: Since you want a very cheap gaming rig. I don't think you want to max out games since you're looking for cheaper ones. If you game at 1080p, this will play all games but at low settings. But very smooth.
PCPartPicker part list:
http://pcpartpicker.com/p/2nDKW
Price breakdown by merchant: http://pcpartpicker.com/p/2nDKW/by_merchant/
Benchmarks: http://pcpartpicker.com/p/2nDKW/benchmarks/
CPU: AMD A10-6800K 4.1GHz Quad-Core Processor ($119.99 @ TigerDirect)
Motherboard: MSI FM2-A75MA-E35 Micro ATX FM2 Motherboard ($59.99 @ Newegg)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1866 Memory ($79.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($59.98 @ OutletPC)
Case: Thermaltake VL80001W2Z ATX Mid Tower Case ($24.99 @ Microcenter)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply ($59.99 @ NCIX US)
Optical Drive: Samsung SH-224DB/BEBE DVD/CD Writer ($14.99 @ Newegg)
Total: $419.92
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-12-21 02:13 EST-0500)