Hello Dalehurd,
an enterprise may require a BGP AS number only when it is truly multihomed, that means it has more then one ISPs that connects it to the BIG internet.
Usually this means the enterprise has also its own IP address space received from a RIR (ARIN, RIPE, LACNIC and so on)
In last years with the introduction of 4 bytes AS number, the problem of lack of BGP AS numbers have been resolved. But I think the rules are still those requiring multihoming.
So I would expect new applicants to receive a 4 byte AS number nowdays.
>> From an Education Perspective. Would a Large University 15K+ students have their own AS?
Some universities may have their own AS number for historical reasons they were the first users of the beginning of Internet.
In general, see for example with RIPE the european RIR the policies for asking an AS are reported here:
https://www.ripe.net/manage-ips-and-asns/as-numbers
Hope to help
Giuseppe