Which Cable Makes the Biggest Impact?

In my limited experience, the current rank from largest to smallest difference.

1. Power cord

2. Speaker cables

3. Interconnects

4. Digital cables (USB)

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.