What is the largest value not attained by coins worth $\$0.25$ and $\$0.29$? (Frobenius Coin Problem)

It’s a well-known result of Sylvester that if $a$ and $b$ are relatively prime positive integers, the largest integer that cannot be written in the form $ma+nb$ for some non-negative integers $m$ and $n$ is $ab-a-b$. (He also showed that exactly half of the $(a-1)(b-1)$ integers $0,1,\dots,ab-a-b$ cannot be so written.) For your problem take $a=25$ and $b=29$ and work in cents.

This PDF gives a lot more information on the Frobenius coin problem, including a pretty accessible development of the first part of Sylvester’s result. (The later parts of the PDF are considerably more advanced.)