So, I have a HP desktop w/ i7-3770 Processor quad-core processor, and terrible graphics card that came with it. I now want to make a large upgrade to play games like bf3 at very high settings, and to play games in the future with high settings. I have decided to either get the 7870, or the 670, or even wait for the 660ti. Will my motherboard bottleneck either of these cards?
I see that you fixed the link, and yep, you're free to use ANY card that you want without fear of any bottlenecks (it's even a PCIe 3.0 slot with that 3770 installed).
The main issue would never really have been the motherboard anyway (motherboards themselves rarely the root of any bottlenecks). The main problem is being sure the the case of that HP system is large enough to fit the card that you want to install.
I honestly don't know. Most good GPU's are in the 9-10" inch range, but depending on how long the drive bays are (how far they protrude into the inside of the case), a 9-10"+ card may not fit.
You'll need to measure the distance between the back of the case to the drive bays to see how much room you have. You really need to have around 10.5-11" of clearance for a 10" long card. That means the drive bays will need to be around 5.5-6" long.