Fixing a (possibility) incorrectly wired 4 way light
Wow, this is a tangle.  Here's what's really going on here.  

There are 2 lights!

Your photo doesn't document which (non-visible) terminal on the 3-way is common. But we can figure it out for 2 reasons: a) we can see one traveler, and both travelers are always in the same cable.  And b) the upper right has 2 wires, and there would be no earthly reason for another cable to tap one traveler.  So confidence 100% on that.  

When the non-traveler in the cable is white, it might be neutral. But this one doesn't go to  the one neutral bundle. This one joins two other blacks.  That doesn't tell us whether it's always-hot or switched-hot.   But given that there are 2 wires on the 3-way common, that leaves only two possibilities: 


The 3-way common hosts always-hot and the other wire carries power onward to some other point of use.  However, that means the other switch is "slaved" to the 3-way system, and only can be turned on if the 3-way system is on.  That might make sense for an "add-on" light, but is insane for a stairway vs porch light. 
The white wire is always-hot, and the suppply hot (the black wire) serves a) the 1-way porch light via the pigtail, and b) the 3-way stairway complex via this white wire.  That feels more correct. However, the 2 wires on the 3-way common mean that this is being distributed to 2 lights, both from here.  


The second one makes sense to me, so I'm going for that.  Here, I've installed "colored tape" and a tracer line on the appropriate wires to designate their function (I forgot the switched-hot for the porch light; tag that blue).  


Black for always-hot  
Red for switched-hot 
Blue for "alternate" switched-hot (i.e. second switch) 
Yellow for both travelers (no need to distinguish them). 
White for neutral and when a white is used for non-neutral, marking it is mandatory.




If testing proves out this guess, I strongly recommend physically marking the wires as such.  Given the complexity of this circuit, it will save you or the next person much head-scratching later. 

Really, wires should be marked the same on both ends, but I don't see a need to get out the fancy ladder to access stairwell lights just to mark the hot red; it should be obvious to anyone that a lamp's hot wire would be switched-hot.  The 4-way, on the other hand...

Colors mean nothing in 3/4 way circuits

So what we know, is if black and red are travelers on this cable, then that must also be true for this cable - this cable only - at the 4-way switch.  So if  you can measure the 4-way and figure out which terminals change when this switch is thrown (i.e. do not change when the 4-way is thrown), you can mark those 2 travelers yellow also.  You MUST mark that cable's white black. (red is legal, if it proves out to be a switched-hot after all, but you can't leave it white). 

Now with half the 4-way switch's wires identified, the other 2 wires from the other cable must also be travelers.  Mark those yellow as well.  (as long as you can see how they're grouped in cables, introducing another color doesn't really benefit; travelers from one  cable go to the black screws, the travelers from the other cable go to the brass. Not hard.) 

The remaining wire in the other /3 cable must be whatever this remarked-white proves to be. (presumably, always-hot).  Note that aside from marking reused white wires, Code says that if neutral is absent but always-hot is present, white must be used for it - that way it always tests as hot, leaving no doubt it is not a neutral. 

Well, that was hard.  If you remark wires as I advise, it won't be hard the next time. 

Y'know, old Code used to say you could skip marking the neutral if the usage was obvious - Code meant "like in an old style switch loop, 2 wires, no question".  There is nothing obvious about this setup!!! But obviously the electrician got in the "habit" of just never marking white wires ever, and this is the result. 

By the way, marking is mandatory.  Marking for different kinds of hot (black red yellow) is entirely optional. For all Code cares, they could all be black.  I disagree, sincerely: I own 11 different colors of THHN wire, all #12 stranded. Taste the rainbow! 

Which of these wires doesn't belong?

Speaking of colors, I note a brown wire in there.  Nope. Nope. Nope.  That is thermostat or doorbell wire, and it does not belong with the others.  I gather you're bumping this junction box out to a 3-gang box.  Make sure to install a divider to separate the low voltage in the 3rd gang from the mains voltage in the first 2 gangs.  You cannot mix low voltage and mains wires.


Unless all the low-voltage wiring and equipment is entirely contained within Class I wiring methods; i.e. if you used #14 Romex instead of that thermostat cable, and the doorbell was mains rated and housed in a Class I junction box, and so was the doorbell.  The reason for this restriction is a low voltage wire coming in contact with mains and being "lit up" at mains voltage, causing wire meltdowns, shocks and fires in the equipment not rated for contact with mains.  


Most plastic box makers offer some sort of divider; however I'm a big fan of steel boxes because they will dissipate arcing heat instead of burning through. You can get dividers for some of those too. 

I don't know how you removed the old box, but you will need to disconnect the wires so the cables can properly enter the new box on a cable clamp. That wire marking will really pay off!