What do you call a cardinal $\kappa$ that is a limit of $\kappa$-many cardinals? For instance, $\omega$ is the limit of $\omega$-many cardinals.  But of course $\omega_1$ is not the limit of $\omega_1$-many cardinals.
1) Are there cardinals other than $\omega$ with this property? Are they easy to find and arbitrarily large?
2) If so, what do you call them?
 A: The aleph function $\alpha\mapsto\aleph_\alpha$ is the increasing enumeration of the infinite cardinals. An infinite cardinal $\kappa$ has precisely $\kappa$ cardinals below iff $\kappa$ is either $\aleph_0=\omega$, or else $\kappa=\aleph_\kappa$, that is, what you are after are precisely the fixed-points of the aleph function (and that's how these cardinals are typically identified, as ``fixed points''). 
Recall that a function $f\colon\mathsf{ORD}\to\mathsf{ORD}$ is normal iff $f$ is strictly increasing and continuous (note the last condition simply says that $f(\lambda)=\sup_{\alpha<\lambda}f(\alpha)$ for $\lambda$ limit). For any normal function there are arbitrarily large ordinals $\alpha$ such that $f(\alpha)=\alpha$: Given any $\tau$, start with $\beta_0>\tau$, and define $\beta_{n+1}=f(\beta_n)$ for all $n\in\omega$. Since $f$ is strictly increasing, we have that $\tau<\beta_0\le\beta_1\le\beta_2\le\dots$ and, letting $\beta_\omega=\sup_n\beta_n$, then continuity ensures that $f(\beta_\omega)=\beta_\omega$.
Although the ordinal $\beta_\omega$ as above is typically of cofinality $\omega$ (unless it happens that $\beta_0$ is already a fixed point of $f$, and of uncountable cofinality), it is in fact the case that $f$ admits fixed points of arbitrarily large cofinality: Simply note that continuity of $f$ ensures that the increasing enumeration of the fixed points of $f$ is also a normal function. For any limit ordinal $\tau$, it follows that the $\tau$-th such fixed point has cofinality $\mathrm{cf}(\tau)$.
The result that there are arbitrarily large fixed points (of the aleph function), of arbitrarily large cofinality, follows at once noticing that the aleph function is normal. On the other hand, we cannot quite ensure in $\mathsf{ZFC}$ that there are regular fixed points, as these are precisely the (weakly) inaccessible cardinals. 
