What's The Cheapest Gaming Desktop Setup to run any game smoothly?

Look this is kinda an opinion question more than any thing its kinda hard to come up with some thing with out a budgte and more info. Things we need to know include do you have any parts you can reuse. Do you have a monitor, mouse and keyboard? do you need to purchase a copy of windows? What resolution are you looking to play at. Any way here is what i would suggest for a budget gaming pc.

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: AMD Athlon II X4 750K 3.4GHz Quad-Core Processor ($79.99 @ Amazon)
Motherboard: ASRock FM2A75 Pro4-M Micro ATX FM2 Motherboard ($74.99 @ Amazon)
Memory: Crucial Ballistix Sport XT 8GB (2 x 4GB) DDR3-1600 Memory ($54.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($59.99 @ NCIX US)
Video Card: PowerColor Radeon HD 7850 1GB Video Card ($129.99 @ Newegg)
Case: Rosewill REDBONE ATX Mid Tower Case ($39.99 @ Newegg)
Power Supply: Corsair Builder 430W 80 PLUS Certified ATX12V Power Supply ($44.99 @ Newegg)
Optical Drive: Samsung SH-224DB/BEBE DVD/CD Writer ($14.99 @ Newegg)
Operating System: Microsoft Windows 8 (OEM) (64-bit) ($89.74 @ Amazon)
Total: $589.66
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-07-24 23:13 EDT-0400)

This build goes past what i would consider minimally acceptable performance. I allocated more cash in the name of a better price to performance ratio and an over all nicer machine. Maxime506's first build is in line with what i would consider bare minimum when comparing our builds keep in mind mine includes a copy of windows which adds $90 once thats taken into account the price difference is only $130. I'm going to break down what that extra 130$ gets you item by item.

The Athlon II X4 645 suggested by maxime is almost 3 years old an extra 5 bucks nets you the Athlon II X4 750K which is much newer has twice the L2 cache, higher clock rate and is multiplier unlocked to allow for overclocking.

The cheapest FM2 boards are about $50 and an extra 25$ gets you usb 3.0, 4 memory slots vs 2 for the cheap boards, e-sata, and better on board audio.

$25 buys you and extra 4gb of memory and gets you a dual channel kit doubling data transfer rates. 4gb is ok but 8gb is much more future proof and i find i make use of it on my mahcine. If you don't want the extra ram at least go for a dual channel 4gb kit its only 5 bucks more.

An extra $10 doubles the hard drive capacity bringing it up to 1TB.

The 7850 is where most of the performance increase comes from an extra $35 gets you a 40% to 50% increase in gpu performance the 7850 1gb provides exceptional value for your money.

$10 buys you a case i feel is better looking and has better cooling.

Same power supply the corsair builder 430W is semi modular, has enough power to run every thing here and won't catch fire.

As far as optical drives go just get what ever is cheap.

As far as the OS goes a gaming pc needs windows and both 7 and 8 go for about 90 dollars i'm suggesting windows 8 it boots faster and will support direct X 11.2.

If your budget is really tight i would take just the Athlon II 750k, the 7850 and a dual channel 4gb kit from this its only 50 bucks over maxime's build. That $50 buys a lot of performance and would give you a good 1080p gaming experience.