Appears brand new nozzle is jammed  

Grub screws

If the extruder grub screws work loose the you will have the symptoms you describe. A quick test would be to open the extruder door, heat up the hot end, then manually push the filament through.  If it comes out the nozzle, check the grub screws. If does not, there is probably a jam

jwv

no grub screws

The grub screws are still tight, and the gear isn't slipping as far as I can see when I spin it.

Check for obstruction

@jwvaughn's tip is a good one:

1. Heat the nozzle up to printing temperature.
2. Open the extruder door.
3. Manually feed filament down through the top of the extrder, past the open extruder Bondtech gear, down into the PTFE tubing, and into the hotend. If your filament path is clear, you should be able to push filament out the nozzle. If not, you may have a partial clog or obstruction.

question

When I have the nozzle removed the filament goes through fine, but when the nozzle is added it jams. This is a brand new nozzle that's never printed before.

Could I still have a clog somehow?

can you see light through the brand new nozzle?

what diameter is the filament exit hole in the nozzle?

if it's 0.4mm, have you tried probing the nozzle from below, with the acupuncture needle, whilst the nozzle is hot?
have you checked the clogged nozzle procedure?
https://help.prusa3d.com/en/article/clogged-nozzle-hotend-mk3s-mk2-5s_2008

Have you tried a cold pull? 

https://help.prusa3d.com/en/article/cold-pull-mk3s-mk2-5s_2075/

regards Joan

prurging

Does jam only when printing or also when plain purging?

Posted by: @flyingpurplecow

When I have the nozzle removed the filament goes through fine, but when the nozzle is added it jams. This is a brand new nozzle that's never printed before.

Could I still have a clog somehow?

waiting on my kit

I am waiting for my kit to come with the needles in it. Won't be able to attempt this until at least Sunday 🙁

purged

Only jams when the nozzle is on there. When the nozzle isn't there it goes through just fine.

clarify

Can you purge with the nozzle on?  How tight is your idler screw?  

Posted by: @flyingpurplecow

Only jams when the nozzle is on there. When the nozzle isn't there it goes through just fine.

Can't purge with nozzle

No, I can't purge with the nozzle on, and I had the idler set correctly when it was printing before with the original nozzle.

It appeared to jam up while printing, and filament wouldn't go through unless the nozzle wasn't there. I tried a new nozzle, while waiting for my cleaning kit to come in, thinking that that might work in the meantime. It still jammed. I can purge without any nozzle, though. 

Jammed again

I cleaned the nozzle with the needle, and it appeared to start working. It was purging just fine. I started a print, and the purge line worked, but then failed when it started printing. Back to what it was doing with the clicking and everything. 

Filament goes through without nozzle. Nozzle looks okay and you tried different ones. I assume you're printing at the correct temperature for the filament. Leaves pretty much only the idler screw as the culprit. Don't see what else it could be. I'd try tightening it a bit.

jams

Idler is most likely, also look at the Z height.  It could be that it is too low.  Too low can cause jams.  

Posted by: @fuchsr

Filament goes through without nozzle. Nozzle looks okay and you tried different ones. I assume you're printing at the correct temperature for the filament. Leaves pretty much only the idler screw as the culprit. Don't see what else it could be. I'd try tightening it a bit.

look at the Z height

OP said he can't purge with the nozzle on, so I took that as the issue being independent of Z height. Otherwise you're of course right. If the nozzle drags in filament, clogs are just waiting to happen.

idler screw

I have tried both loosening and tightening the idler screw. The tighter it gets, the louder the clicking. 

I'm starting to think I need to bite the bullet and disassemble the extruder. Wanted to have to avoid this given how difficult assembly was.

Keep us posted. My money was on the extruder gear assembly/idler screw, but I read the whole thread again, and at one point the OP said he could push the filament through by hand without the nozzle but not with the nozzle on, which appears to rule out the extruder ☹️ Now I'm baffled.

Heat Creep?

Just a thought: have you checked the hotend fan?  Heat creep can cause similar sorts of problems, particularly with PL.

This may not be the problem but it's worth checking.

options

It could also be the idler screw, z offset too low, or heat creep.  

Perhaps filament

I read a post some time ago where the nozzle kept clogging and the user was out of options.  He switched to  a new roll of filament and things went back to normal.  His theory was that at some point the offending roll was contaminated with particulate during manufacturing that was big enough to clog the nozzle.  The chance that a roll of filament would get contaminated (and not necessarily the whole spool) is remote but possible so, maybe when all else fails to fix the problem, try another spool.

Just for info, there was a more recent case where the spool was labeled wrong and was loaded with a filament requiring a higher temperature. You did not mention switching to a new roll but I thought I would mention this just in case.

jwv