How large could a second moon of Earth be? Question in the title. Let's say that I have a lot of garbage- mined-out asteroids, spent nuclear fuel, satellite debris, and just general trash. Also safe ways to get these into Earth orbit. Instead of sending this into the Sun, Jupiter, or out of the Solar System's plane, what if I took some random quasi-satellite, put it a bit behind the Moon on its orbital path, and started piling garbage on it? 

How large could it get before bad things start happen to Earth? For example, tides drowning the coasts. I'm not going to consider collisions, because of (insert megacorp here) Moon Tethers™.
Your question doesn't work because it's based on a false premise - that you can have a stable 2nd moon in orbit. You can't. If you accept the fact that the 2nd moon won't be stable, the size it becomes dangerous is the size that you wouldn't want to fall to Earth - if it's garbage and loosely held together, 100-200 feet in diameter would probably be safe enough and make a big flash and a lot of noise if it hits the Earth, but not cause too much damage.

Space garbage disposal is pretty easy - because there's so much empty space. In densely packed regions like low Earth orbit with lots of satellites, you'd want to avoid dumping garbage, but in the vastness of the Earth's Hill sphere, letting garbage just float in empty space wouldn't be all that dangerous, unless there was a lot of it. If there's enough manufacturing that you can't let garbage float, or if there's toxic minerals like lead or plutonium that you don't want to let fall to Earth, then creating a garbage dump in space makes sense, but it'd be way smarter to just use the Moon as a garbage dump than to build a 2nd one.

If the Earth-Moon system could hold an additional moon in a stable orbit, then your question on how massive could be answered, but there's no practical reason and no benefit towards creating a garbage moon, it would have an unstable orbit and require periodic re-adjusting or it would be dangerous. Just make a garbage dump on the moon.  It's way smarter.  

Would the L4 and L5 Lagrange points be unstable?

Doing a bit more research, it seems that Earth-Moon L4 and L5 are stable. NASA did some research into it. Perhaps that makes my answer not correct to this question.  An L5 garbage moon would be possible. - See @Andy256 answer here.

As noted in the comments, a moon-sized object is a lot of material. The Moon is 20 times the mass of the asteroid belt. The number of comets and asteroids that would need to be captured, mined and discarded into the L5 orbit to create a 2nd moon is huge. We're not likely to create anything but a tiny moon out of captured objects because the Moon is comparatively enormous. I rather like the idea of using L5 as a place to capture, place and mine asteroids.

There's still a size limit though. If it grows to a few times bigger than the Moon, it's no longer the Moon's trojan, the Moon becomes its trojan and there's a mass limit for a stable trojan orbit, perhaps about 2 lunar masses, after which, the Moon's orbit would become unstable - and could be a world-ending scenario.   

(old update below)

That's a tough call. The Earth-Moon system is far from ideal for stable L4/L5 Lagrange points because the Moon's orbit wobbles due to solar tugging. The mass ratio of 81 to 1 is large enough, with about 26 to 1 being the minimum required for stable L4/L5 saddle points, but the eccentricity and lunar apsidal precession could be problematic for long term stability.
I'm not smart enough to say for certain whether they would be stable or not.  Perhaps another question, or it also fits here if someone else wants to answer.