I need around 120 digital outputs for a project, how would you suggest I do this?

You might check out Kevin Darrah's projects on Youtube with LEDs and shift registers.  https://youtu.be/zS5oZPJVh_o  He also has his own webpage with some serious looking proto boards.  http://www.kevindarrah.com/?page_id=1618

I don't know this person, nor have I ever used any of his boards, but the project videos he has out there are definitely interesting, add in the fact he gives step by step instructions and code as well and you might enjoy remixing some of that into your project.

This is a very similar discussion.

My 3rd big Halloween Project - an actual nightmare

One simple way to do this is using addressable LEDs (such as WS2811, WS2812) - one serial output can easily control 144 LEDs individually, including colour and intensity. You can buy them in flexible strips if that form factor works for your application.

To me the easiest way (although probably the least professional) would be to get cheap a couple of cheap microcontrollers (arduino uno or whatever) and connect them to one master controller using spi or i2c.

Otherwise, you could use port expansion chips like these NXP I2C Agile I/O Peripheral Expanders :: NXP Semiconductors, but as far as I have found, microcontrollers are cheaper lol.

@John_kutzschebauch :   Thank you for that great vid, I had seen some RGB cubes before and always wondered what method they used and if they had some complicated system with LEDs in series but that video was really helpful. I don't think I will use the same way as he used though as it seems very complex, I had seen something using the same chip beforehand. Thanks a lot for the suggestions, I will keep them in mind possibly for a later project to learn about that chip.

@Sarünas Staigis : Thanks for the comment! As for the two microcontroller idea, I had touched on using two arduino mega type ones but I would need more than two of those and I am not sure at all how the code would work. Another problem that I am still uncertain about is that I need it to fit into a relatively small space (not sure what size yet) and arduinos are chunky for what they are. Thank you again for the ideas, I may use the master + slave system with PIs one day for something. I am glad you made your comment as it clarified that my original idea was in fact possible and that I wasn't being crazy! Thank you again.

You could also use 32bit expander: Arduino Interface PCA9555 GPIO Expander

And this is probably the cleanest solution: ams’ AS1130 is the smallest LED dot-matrix driver for 132 LEDs

This person has a 3 part series on making a 15X10 led matrix that I found entertaining and informative https://www.youtube.com/watch?v=8pFcZ-ijrDY. He uses through hole leds instead of the strips and explains the difference between using RGB Leds and addressable Leds for larger scale projects which i enjoyed. I hope this is helpful.