Non-bouncy numbers

Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, $134468$.

Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, $66420$.

We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, $155349$.

As n increases, the proportion of bouncy numbers below n increases such that there are only $12951$ numbers below one-million that are not bouncy and only $277032$ non-bouncy numbers below $10^{10}$.

How many numbers below a googol ($10^{100}$) are not bouncy?