Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any integer base system because integer base systems are all about rationals while pi is inherently irrational. It's a no-brainer why all representations of pi have been infinite. It's because we've chosen natural numbers to study this more complex phenomenon known as pi. Base systems are all about strings of natural numbers. Pi goes beyound natural numbers.So how about we change the system? Is there any system you know of where pi can be represented finitely?
UPDATE More specifically, Do you know of any number system (besides base pi, base root pi, etc) in which pi can be represented finitely? It need not be a base-system.
Also, the current answer by @HenningMakholm is technically a finite representation of pi, but I don't think it qualifies as a number system.