These days, the easiest way (in Windows) is to run the charmap program - see the previous post for the location. This little utility allows you to copy and paste any character into your application, including a bunch of math symbols. I don't know what, if any, is the limit these days but the sets have heaps and heaps of characters.
These days, most modern computer systems use use Unicode (or ISO/IEC 10646, which is essentially the same thing) for character coding. In its fullest form, this uses 32 bits to encode characters. This could permit up to 4294967296 characters in principle, although for technical reasons not all code positions are available for distinct characters. The number of characters for which codes are defined is still growing, but few, if any charaters are yet assigned codes that require more than 16 bits. That is still 65536 possible characters. Whether your PC or MAC can display a particular code is another question, with the answer depending on the fonts available and the characters supported by any particular font.
By the way, character sets for which codes have recently been defined include Game Tiles, Egyptian Heiroglyphics, Phoenician and Phags-pa. (I don't know what that last one is either.)