It is known that the moduli space $\overline{M}\_{g}$ of genus $g$ curves is of general type for $g\geq 24$. 


By Theorem 2.4 of 


*Logan, Adam The Kodaira dimension of moduli spaces of curves with marked points. Amer. J. Math. 125 (2003), no. 1, 105–138.*


all but finitely many of the $\overline{M}\_{g,n}$ with $g>3$ are of general type. Furthermore in Theorem 5.1 of the same paper for any $4\leq g\leq 23$ an $n$ such that $\overline{M}\_{g,n}$ is of general type is determined.


**Is there an inequality in terms of $g,n$ which implies that $\overline{M}\_{g,n}$ is of general type ?** 



Kodaira dimension of the moduli space of curves
So it seems your question is: if we know that $\overline M\_{g,n}$ is of general type, is the same true for $\overline M\_{g,n'}$ with $n' \geq n$? The answer is yes, and is a special case of the main theorem of: *J. Kollár, Subadditivity of the Kodaira dimension : fibers of general type*.


As far as I know there are no explicit counterexamples to the stronger statement that if $\overline M\_{g,n}$ is of general type, then the same true for $\overline M\_{g',n}$ with $g' \geq g$ and $n' \geq n$, but it is almost certainly false: for instance, Farkas has proved that $\overline M\_{22}$ is of general type and also that $\kappa(\overline M\_{23}) \geq 2$, which he however conjectured to be sharp. Also note that Logan's function $f(g)$ such that $\overline M\_{g,n}$ is of general type for $n \geq f(g)$ is not monotone in $g$.


