More formally, find an asymptotic for $N\to\infty$ of $\frac{\sum_{1\le k\le N} M(k)}{N}$ where $M(p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}) = d_1+d_2+\cdots+d_k$
For example, $M(24) = M(2^3\cdot3) = 4$.
More formally, find an asymptotic for $N\to\infty$ of $\frac{\sum_{1\le k\le N} M(k)}{N}$ where $M(p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}) = d_1+d_2+\cdots+d_k$
For example, $M(24) = M(2^3\cdot3) = 4$.
Check out equation (3) here. Unsurprisingly, the first term in the series is the same as the one for the distinct divisor function $\omega(n)$, namely $\log \log n$.
equation (20) http://mathworld.wolfram.com/PrimeFactor.html gives the exact answer as $\log \log N +B$ for some constant B. The link gives reference to what the value of B is.