Can you apply the distributive law forever?

(Lipton)

You have an arithmetic expression that adds and multiplies some variables.

For example $(x + y) * (z + a * (b + c ))$ .

You repeatedly reply the distributive law to sub-expressions of the expression.
For example, you could replace $a*(b + c)$ with $(a * b + b * c)$.

Prove that you cannot do this infinitely many times.
Bound the number of times you can, given an expression with $N$ variables.