Which Cable Makes the Biggest Impact?

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.

An interesting question without a definitive answer, but I've learned a few things since jumping into the shallow end of the pool more than a few decades ago. Things have changed since then.

About ten years ago I began playing more with cables in a limited way. The effects were limited, but real. Still about knee deep in the pool at this point.

In the last few years I reached the deep end and began serious cable swapping. I highly suspect my conclusions are unique to our times and what I've assembled. Until I had some serious gear, I can't say I could have fairly evaluated any cable. Now the differences are readily apparent.

Being all digital in the front end now, any cable or device that keeps out or filters out RFI and EMI from the digital processing equipment has the largest effect. This is power cords, conditioners, and digital cables. Without these, the sound is gray, compressed, rounded, and muddied. The stage is narrow and shallow. This can't be fixed downstream. We are awash in RFI these days.

But to assign a ranking still depends on how well the components are designed and built. So generally...

1. Power cords, especially those with noise reduction capability for source equipment.

2. USB/Digital cables (of course with digital sources)

3. Speaker cables (part of the all important amp and speaker pairing)

4. Interconnect (preamp to amp, then lastly source to preamp)

Twenty years ago I would have said the interconnect between source and pre was most important. But now my ears say otherwise. Times have changed.

It's still very true that the last weak link kills the party in a high quality system. If short one power cord, the whole system will sound like whatever the bad one is. And it will drive you nuts.