jmc

There is $1$ two in the one-digit numbers. The number of two-digit numbers with a two in the tens place is $10$ and the number with a two in the ones place is $9$. Thus the digit two is used $10+9=19$ times for the two digit numbers. Now, Pat Peano only has $22-1-19=2$ remaining twos. You must subtract 1 because 22 is counted twice. The last numbers with a two that he can write are $102$ and $112$. He can continue numbering the last couple pages without a two until $120$, with the last number he writes being $\boxed{119}$.