Is there any limitation on the number of times I install a custom-ROM to my android device? How many times can I install a custom ROM to my android device? Can I do this an infinite number of times? The word ROM (Read-Only-Memory) makes me think that there must be some limitation to the number of times one can do this.
ROM refers to two things


The memory (non volatile) in the phone 
The software or Operating System (OS) that is run on the phone and resides or is "flashed" on the memory portion referred above


In common Android usage when you say ROM, you are usually referring to the second such as stock ROM (one that came with the phone), custom ROM (one that is developed) which you flash

You can flash or install a ROM as many times but certainly not infinite number of times- something in the hardware / memory is bound to break at some point of time ! But yes, theoretically you can  

Related What is the difference between: Rooting, Jailbreak, ROM, Mod, etc.?