Identifying strange 3 way / 4 way switch wires
THIS IS A GUESS - This Explanation and diagrams must be confirmed with a tester.

I added this as a comment but I'll go ahead and make an answer so I can post 2 drawings.  This is what I think you have going on, I think the original plan was for the circuit to have just 2 switches as classic 3-way, like this:

(Excuse my Paint abilities, NOTE: GROUNDS NOT SHOWN
 

The Next day, Next week or Next year someone decided to add a 3rd switch, the load end of the original 3-way circuit became the 4-way, like this:

This is one explanation for what I see in your pics, but I can't see everything needed. THIS IS A GUESS AT THIS POINT, using these diagrams as a guide you need to test and confirm the accuracy.

The clues that suggest this explaination to me are the white wire attaching to the common screw in your second pic, and in the first pic a white wire nutted to a pair of black wires. The guess is that's the same white wire. A good first test would be to check for voltage where the white joins the 2 blacks, if my theory is correct that will but hot when the circuit is on and not hot when the circuit is off.