I was given the rules:
Parts with Amazon Prime shipping only.
Must choose the absolute cheapest parts - no picking something "a lot better for just $1 more". Must go at least 5 pages deep in search results.
Purchase must include 3 full sets of props and a battery.
Don't need to buy a charger or Tx. Can use an Rx typically bundled with the Tx; any other models would have to be purchased.
Must be capable of carrying a camera on a gimbal (don't need to include either item in the build though).
Any assembly materials not considered common household items must be purchased. (we consider zip ties to be common household items, along with soldering irons and solder)
Here's what I ended up with:
Frame - $17.49 - F450 clone. This actually should've been like $13.50, but the item sold out while I was putting together the build. This was the next cheapest.
ESCs, Motors, 2 sets of props - $74.95 - The included props are what really drove this option home. I had a few other ESC + motor sets picked out for around the same price, but since Amazon sellers overcharge so much for props, having 2 sets of props included in this kit was a huge boost.
Flight Controller - $24.90 - The CC3D seems to be the prolific flight controller available from Prime sellers. I had hoped for a Naze, but oh well - at least I didn't get stuck with a Multiwii or KK.
Landing Skids - $16.58 - Technically, with the "absolute cheapest" rule, this should've been these $7.01 legs, but I decided with the camera/gimbal requirement, it was important to get landing gear that weren't prone to flipping your shit.
Battery - $22.99 - Yeah, not a lot of great battery selection on Amazon. Since we didn't have any rules in place about battery size, I just chose something comfortably around $20.
One more set of props - $7.50 - They don't perfectly match the included props, but are close enough that it might not matter if I end up needing to use them at the same time.
Silicone Wire & Dean's plugs - $15.99 + $6.50 - Not counting these in the build price, as I was mostly stocking up for myself - but one side of a T connector and about 8" total of wire did get used in my build.
I used an old Turnigy 9X8C Rx (the module included with the 9X Tx bundle). Mounted the Rx and FC using double sided foam tape meant for holding hooks on a wall, and obviously the zip ties for the ESCs/arms.
All in all took about 3 hours to build - including a ton of time wasted disassembling and re-assembling the entire frame (incl motors) to rotate it to put the yellow in front, and making a Deans <-> XT-60 adapter dongle so I could test using the only batteries I have handy (which I also had to make twice since I'm an idiot and used the wrong side of a Dean's plug on the first one).
EDIT: For anyone considering a less crappy version of this build concept
As brought up in this comment. I actually wouldn't really change much. Get a frame in the color you like (I saw red/white, white/yellow, black/white, black/red, and red/white deadcat). Swap the motor and esc bundle for this fantastic combo for $55 more (bonus: bullet connected motors/escs, reduces soldering requirements to 10 ridiculously easy pads + a power adapter). Buy a BEC, lots of extra props, and a set of prop nuts (that bundles included props are self-locking). Maybe order an APM off a domestic ebay seller instead of a CC3D from Amazon.
EDIT 2: Video & plug
Took it outside for a test flight today. Nothing exciting - there are too many trees nearby for me to risk going high with an untested quadcopter; plus as you can hear, my transmitter battery was a little low. I'll update the imgur post with a screenshot of my PIDs later today, and hopefully get out to a park this weekend for some real flying!
Also, be sure to check out /r/multicopterbuilds/ - a new subreddit started as a result of this thread!