jmc

By the change-of-base formula, $$\begin{array}{rl}\frac{1}{{\log }_{15}2+1}+\frac{1}{{\log }_{10}3+1}+\frac{1}{{\log }_{6}5+1}& =\frac{1}{\frac{\log 2}{\log 15}+1}+\frac{1}{\frac{\log 3}{\log 10}+1}+\frac{1}{\frac{\log 5}{\log 6}+1}\\ & =\frac{\log 15}{\log 2+\log 15}+\frac{\log 10}{\log 3+\log 10}+\frac{\log 6}{\log 5+\log 6}\\ & =\frac{\log 15}{\log 30}+\frac{\log 10}{\log 30}+\frac{\log 6}{\log 30}\\ & =\frac{\log 15+\log 10+\log 6}{\log 30}\\ & =\frac{\log 900}{\log 30}=\frac{2\log 30}{\log 30}=\boxed{2}.\end{array}$$