(You can use this fact to distinguish between multiple connected Arduinos and have them do different things. For instance, I have one Arduino that is “/dev/tty.usbserial-A4001lZf” and another Arduino that is “/dev/tty.usbserial-A7006xIM”)

When controlling LEDs there are usually two things you have to worry about:
- power draw — how much current needed for your LED array
- addressing — how to control a specific LED in your array

By saying external power source, you get around the power draw issues. An Arduino on a USB port can supply about 500mA of current. A standard LED draws max about 20mA of current. So you can have 500/20 = 25 LEDs off a single USB-powered Arduino. Of course, the Arduino chip itself can’t supply enough power directly to do all 25, you’d need some external driver transistors to help out. Or use LED driver chips.

As for addressing, an Arduino has 13 digital I/O pins. If you set those up as a matrix of 6 pins by 7 pins, you could address 42 LEDs with just the Arduino. But since you include LED driver chips and shift registers in the mix, those chips can do all the hard work and you just talk to them via multi-node serial protocols like I2C or SPI. There are LED driver chips which can address 24 LEDs at a time and are meant to be bussed together in a long string. So you could have maybe 100 of these chips and be able to have 240 individually-addressable LEDs.

But now you have to start worrying about logical addressing. If you think about what an Arduino program to control a few hundred LEDs would look like, you’d probably have a big array in memory, with each array byte is the brightness value of one LED. (many call it a “framebuffer” after what video cards do) A standard Arduino has 1024 bytes of RAM, of which about 256 bytes of this is used by various libraries like Serial. Your program’s non-framebuffer variables take up some of that too. So let’s just say that you have 512 bytes of RAM free for your framebuffer. That means 512 LEDs is your maximum.

Of course, if you don’t need to remember the value for each and every LED (like if you’re doing algorithmic stuff where a “next” LED value can be computed by a small number of “previous” LED values), then your memory requirements can be much smaller than one-byte-per-LED.

If you’re asking your question for more then theoretical curiosity, like say for a particular application, it might be better to say what you’re trying to do. Like most things in technology, there are many options, so the problem needs to be defined better so a solution can be properly fitted to it.