Can we expand numbers on the left to produce irrational numbers?

I suggest you look into p-adic numbers, which is the most well-known way to make sense of "numbers notated with an infinity of digits to the left". But it's not exactly what you're proposing here.

Your idea about monoids is a red herring, I'm afraid. The free monoid contains only finite words, which in this context encode the ordinary finite integers.

And notice that "3.14159265..." is not really syntax. The real decimal expansion of $\pi$ is an infinitary object that can't be written down in full, so "3.14159265..." is either just an evocative hint of the real thing or a practical approximation.

What you find is called a $p$-adic number.

http://en.wikipedia.org/wiki/P-adic_number