How many unique molecules exist? And how many exist only by synthesis?
Polymers make this question more or less unanswerable.  

Consider human chromosome 1, which contains about 249,000,000 base pairs.  There is nothing that says we couldn't order those pairs in any way we like, so for DNA alone, in a quantity that exists in every person on Earth, there exists the possibility for 4249000000 different molecules.  Include things like alternative nucleotides (there are almost surely at least thousands of these that are possible) and longer chains and the number gets even more enormous, and that's only including DNA (and close relatives.)

There's no real way to include all of them, since there have to be an enormous number of ways to put these molecules together that we haven't found yet.

How many exist?  An awful lot.
How many only exist in our minds?  Almost all of them (the universe only contains about 1080 particles, after all.