[abeld]

I think Knuth's up-arrow notation (see http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation) can be considered a newer way of representing numbers, and it continues the progression you describe in that each the older notations are usable for small numbers, but the newer notations are better for large ones: Knuth's up-arrow notation is only really required for truly gigantic numbers and as a result I don't think it would be generally useful. (As generally useful numbers aren't that huge.)

[pmelendez]

I thought Knuth's up-arrow notation was actually an operator rather than a numeral system. Just in the same way that multiplication is a short for sums, the arrow operator would be a short for exponentiation.