Which Cable Makes the Biggest Impact?

Absolutely agree with folks saying cables are complimentary not the key driver for the sound of the system. 
 

My upgrade experience includes upgrading speakers significantly above the grade of my electronics then upgrading my sources, then amplifier. In my case I started with speaker cables and felt it was worth the impact.  And this experience reinforces the recommendations echoed by other…”try it before you buy it if you can”

For me in my system.

Power cords

Speaker cables

Interconnects

Usb cables

Power cords most definitely.

Okay, I'll get in on this. Since the OP's question mainly asks about cables (and not equipment), I'll offer - for my brain and ears:

Biggest Impact(!!): RCA analog interconnects. Number one on my list - hands down 

Runner up: Speaker cables

Lowest impact (again, for my ears and brain): AC power cables

Since I use a mix of balanced and unbalanced analog interconnects; low capacitance, oxygen free copper and heavy shielding (double braid or Reussen), are qualities I seek out. I would also add ethernet cables as other important cables I pay attention to. There's no question in my mind that upgrading a small 3' ethernet cable added more resolution and detail in/to my streaming experience. It's very possible that since I already pay such close attention to my AC, that further upgrading AC cables just never makes a big difference in my audio system.

Let's also keep in mind that audio is very subjective. Everyone's brain and ears are not the same. High performance audio reproduction (and how to attain it) is a learning experience. Everyone has their own priorities. For me, it's a quiet noise floor, with no external EMI or RFI generated pops and clicks. When critically listening, my main source will always be headphones, and headphones can sometimes be like an audio microscope.     

I believe there's no hard rule and it's very system dependent. Especially for power cords: the same power cord can make little to no difference on a given component and produce a "waw!!" moment when connected to another component. 

I make no secret that I use some components that can be considered extremely cheap / an insult to the audiophile world, some might say/ still they work great in their intended purpose. For example, the tiny Trends class D amp that powers the > 500Hz range in my (fully horn loaded) system. In my system, that amp is powered by an eternal linear PSU that costs almost as much as the amp itself; interestingly, using a good power cord on that external PSU was one of the biggest cable upgrade I ever made to my system. Width of stereo image got so much larger  it felt like I inserted some kind of digital processing / effect. Said power cord is almost as expensive as the amp itself - because it's a very cheap amp - so the percentage ratio is ridiculous but the effect is well worth it. It's the result that counts. The same power cord on my preamp made the sound thick and sluggish, but another PC with a very different construction performed miracles there. And so on. 

In any case, while it makes sense to treat the source first, ANY bottleneck in the system will be heard, no matter where it sits in the chain.