Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete.
Does $M_g$ contain an elliptic curve?
The answer is no if $g=2$. In fact, $M_2$ doesn't contain any complete curves.
Note that one can construct complete curves lying in $M_g$ for $g\geq 3$. There are explicit constructions known.
Probably for $g>>0$ and $k$ algebraically closed, the answer is yes.
What if $k$ is a number field?
Best Answer
The easiest way (I know) to see that there are no nonconstant holomorphic maps from a complete elliptic curve $E$ to the stack $M_g$ is to observe that such a map $f$ would lift to a holomorphic map of the universal covers $\tilde{f}: {\mathbb C} \to T_g$, where $T_g$ is the Teichmuller space. The latter is a bounded domain in ${\mathbb C}^{3g-3}$, so Liouville's theorem implies that $f$ is constant.
Edit: Using Kodaira's construction of complete curves in moduli spaces (via ramified coverings of products of curves) one can construct maps from elliptic curves $E$ to the coarse moduli space (of large genus) which are generically 1-1, i.e. 1-1 away from a finite subset of $E$. With more work one can probably get injective maps as well but I do not see sufficient motivation for this.