How far wrong could the Continuum Hypothesis be? I hear it's consistent with ZFC to have
$$ 2^{\aleph_0} = \aleph_n $$
for any $n = 1, 2, 3, \dots $.  How much worse can it get?
More precisely: are there models of ZFC with $2^{\aleph_0} \gt \aleph_n$ for all $n$?   What's the 'world record' when it comes to finding models where $2^{\aleph_0}$ is 'very large' in some sense?  And on the other hand, are there theorems putting interesting bounds on how large $2^{\aleph_0}$ can be?
(I count $2^{\aleph_0} \lt 2^{2^{\aleph_0}}$ as an uninteresting bound.)
According to Wikipedia:

A recent result of Carmi Merimovich shows that, for each $n \ge 1$, it is consistent with ZFC that for each $\kappa$, $2^\kappa$ is the $n$th successor of κ.  On the other hand, László Patai (1930) proved, that if $\gamma$ is an ordinal and for each infinite cardinal $\kappa$, $2^\kappa$ is the $\gamma$th successor of $\kappa$, then $\gamma$ is finite.

However, I'm interested in failures of the Continuum Hypothesis, not the Generalized Continuum Hypothesis.
 A: Solovay proved shortly after Cohen's result on the independence of CH that in any model of set theory $V$, if $\kappa^\omega=\kappa$, then there is a forcing extension in which $2^\omega=\kappa$. The forcing is simply $\text{Add}(\omega,\kappa)$, the forcing to add $\kappa$ many Cohen reals. Thus, the continuum $2^\omega$ can be $\aleph_{\omega+1}$ or $\aleph_{\omega_1}$ or $\aleph_{\omega_1+\omega^2+17}$ or what have you, even weakly inaccessible, if you had such a cardinal available in the ground model. But it cannot be $\aleph_\omega$ or $\aleph_{\omega_1+\omega}$ or any cardinal with cofinality $\omega$, because $(2^\omega)^\omega=2^\omega$ but König's theorem shows $\kappa^{\text{cof}(\kappa)}>\kappa$ for any infinite cardinal $\kappa$. 
In particular, if you start in a model of the GCH, then the continuum can be made to be equal to $\delta^+$ for any infinite successor cardinal in the ground model, and this is cardinal-preserving forcing, so the meaning of the successor function is the same in the ground model as in the extension.
Easton vastly generalized this result to show that if you start in a model of the GCH, and if $E$ is any function defined on the infinite regular cardinals and obeying the following requirements (now called an Easton function)


*

*$E(\kappa)>\kappa$

*Further, $\text{cof}(E(\kappa))>\kappa$

*$\kappa\leq\lambda\to E(\kappa)\leq E(\lambda)$


then $E$ can become the continuum function $\kappa\mapsto 2^\kappa=E(\kappa)$ as computed in a cardinal-preserving forcing extension of the universe. See Easton's theorem. 
The Easton requirements amount to the trivial requirements on the continuum function imposed by the facts that $\kappa\leq\lambda\to 2^\kappa\leq 2^\lambda$ and König's theorem $\text{cof}(2^\kappa)>\kappa$. 
As a result, we can have $2^{\aleph_n}=\aleph_{\omega^2+\omega+p_n}$ where $p_n$ is the $n^{th}$ prime, if we like. There is infinite flexibility, and basically anything is allowed subject to the Easton requirements, which are trivial limitations.
Easton's theorem does not apply at all to singular cardinals, however, and controlling the continuum function at singular cardinals is an extremely subtle issue. 
