jmc

The greatest common divisor of the three terms is $5^3$. Factoring $5^3$ out of each term and using the distributive property gives: $$\begin{array}{rl}15^3+10^4-5^5& =5^3\cdot 3^3+5^3\cdot 5\cdot 2^4-5^3\cdot 5^2\\ & =5^3(3^3+5\cdot 2^4-5^2)\\ & =5^3(27+80-25)\\ & =5^3(82)=2\cdot 5^3\cdot 41.\end{array}$$So the largest prime factor is $\boxed{41}$.