I elaborate a bit on Roy Smith's comment.

One can show that the general curve of genus $g>1$ does not map onto a curve of genus $h>0$ by a dimension count, at least for complex curves.
Let $f\colon C\to D$ be a map of degree $d$ from a curve of genus $g$ to a curve of genus $h>0$ and let $B$ the branch divisor of $f$ (a point $P\in D$ appears in $B$ with multiplicitiy equal to $\sum_{Q\mapsto P}(m_Q-1)$, where $m_Q$ is the order of ramification of $f$ at $Q$.
The Hurwitz formula gives:
$$2g-2=d(2h-2)+\deg B.$$
Let $S$ be the support of $B$ (i.e., $S$ is the set of critical values of $f$) and consider the restricted cover $f_0\colon C\setminus f^{-1}(S)\to D\setminus S$: this is a topological cover of degree $d$ and it determines $f$. There are finitely many such covers, hence the maps $f\colon C\to D$ as above depend on $3h-3+s$ parameters, where $s$ is the cardinality of $s$.
Since $s\le \deg B$, the Hurwitz formula gives:
$$(3g-3)-(3h-3+s)\ge 3(d-1)(h-1)+s/2>0.$$
So the general curve of genus $g>1$ does not have a map of degree $d$ onto a curve of genus $h>0$. Now it is enough to observe that, again by the Hurwitz formula, there are finitely many possibilities for $h$ and $d$.

I don't think that a curve without maps onto curves of positive genus must have a simple Jacobian: if one takes a curve $C$ inside an irregular surface $S$ such that $C$ is an ample divisor of $S$, then there is an injection $Pic^0(S)\to J(C)$, and I do not see why in general $C$ should have a map onto a curve of positive genus. To get an actual counterexample, one could try to look at an abelian surface $S$ with a polarization $L$ of type $(1,3)$ and a curve $C\in |L|$, but I don't know how to show that a general such $C$ does not map onto a curve of positive genus.