What is the maximum number of IPs that can be assigned to a given computer?

I really need to know this for a linux server, but since it also applies to client OSes, I figured the question should be posed here instead of server fault.

According to the replies to this post to the linux-net mailing list,

• At one point, the Linux kernel had a limit of 255 aliases per physical interface device.
• As of Linux kernel version 2.2, there is no (practical) limit, with the second response citing personal experience with having aliased over 5,000 addresses to a single ethernet card.

Of course, binding large numbers of aliases to a single device will impose memory requirements and may have performance implications, so it's entirely possible that your hardware may limit the number of aliases you can add, even if the kernel itself doesn't.

More than you could ever possibly need. Use ip addr add to add addresses to an interface.