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

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.

The maximum would be 4,294,967,294 IP addresses (assuming you're talking about IPV4, you have enough RAM, and we take out one for the broadcast address and one for the network address).