Cheapest laptops that can run Minecraft exceptionally?

You won't find one that low. You can get lucky sometimes with finding one for $650, but if you want a laptop that will run minecraft with no problem. You need to go with something like this.
http://www.newegg.com/...6834231455

I've had the HP pavilion dv6-6c10us laptop for 2 years now (recently broke from a "incident" i won't get into) it has a quad core processor with radeon graphics and seems to run all games on high without little to no fps loss or harship. Price is at 599.99 tho but it streamlines through other apps other than games too.

Try a refurbished apple computer.

$348 Fry's special, says yes.
http://www.frys.com/product/8000684?site=sa:adpages%20page:P1_FRI%20date:050214

http://www.game-debate.com/...quirements

Best Buy Clearance $310
http://www.bestbuy.com/...&cp=3&lp=1
or
http://www.bestbuy.com/...&cp=3&lp=7
A6-5200 Accelerated Processor with AMD Radeon HD 8400
Model: C75D-A7102 available in 17.2" similar to above
Better:
http://www.game-debate.com/...quirements

Also, stronger graphics, weaker CPU does meet ultra also in an HP:
http://www.bestbuy.com/...cp=4&lp=13

and this one is slightly better:
http://www.bestbuy.com/...&cp=5&lp=1
And this one:
http://www.bestbuy.com/...cp=6&lp=15

You need A series AMD anything at OK or better, or a full laptop voltage i3-3xxxM or i3-4xxxM to get HD 4000 or better.

One piece until sold Lenovo Outlet Refurb
http://outlet.lenovo.com/outlet_us/itemdetails/59RF0401/445
Great specs for the price and has a 1 year warranty
A10-5750M HD 8650G
$381 plus tax