Suppressing wages depends on how fast the economy absorbs that "limitless" supply. The problem with immigration at the moment is not wage suppression with completely unskilled day labor valued at $100/day, but with excessive regulation and tax burdens on business that make it more economical to hire temporary illegals than to hire permanent employees. Increasing the labor supply also creates frictional unemployment, but already immigrant employers are hiring other immigrants, so that takes care of itself over time.
I agree about excessive regulation but thats evading the point. My question is how does a limitless supply of illegal labor 'grow' the economic pie? It doesn't - it HAS to stop at some point.