I have to believe this is possible, but I'm stumped. If I have a list:

Dog
Dog
Cat
Cat
Hamster
Dog
Cat
Bird

I want a formula that will search Column A and tell me how many different things are in there... in this instance Formula result = 4

Any ideas?

you can look up help in excel-count unique value among duplicates.

Enter "Count unique values" in Excel help and you will get two different methods. One is to use a filter, but I'll assume you prefer to use a function. Their example is for comparing two lists. In this case we only have one so you can use the following:

=SUM(IF(FREQUENCY(MATCH(B2:B9,B2:B9,0),MATCH(B2:B9 ,B2:B9,0))>0,1))

This assumes your list is in cells B2 to B9. There cannot be any blanks in the list for this to work. Don't ask me how it works. I haven't bothered to figure it out.