Which vector spaces are duals ? Every finite-dimensional vector space is isomorphic to its dual. 
However for an infinite-dimensional vector space $E$ over a field $K$ this is always false since its dual $E^\ast$ is a vector space of strictly larger dimension: $dim_KE \lt dim_K E^\ast $ (dimensions are cardinals of course). This is  a non-trivial statement for which  our friend Andrea Ferretti has given an astonishingly unexpected proof  here. This implies for example that a vector space of countably infinite dimension over a field $K$, like the polynomial ring $K[X]$,  cannot be the dual of any $K$-vector space whatsoever.  
So an infinite-dimensional vector space is not isomorphic to its dual but it could be isomorphic to the dual of another vector space and my question is: which vector spaces are isomorphic to the dual of some other vector space and which are not?
In order to make the question a little more precise, let me remind you of an amazing theorem, ascribed by Jacobson (page 246) to Kaplanski and Erdős: 
The Kaplanski-Erdős theorem : Let $K$ be a field and $E$ an infinite-dimensional  $K$-vector space . Then for the dual $E^\ast$ of $E$ the formula $dim_K (E^\ast) = card  (E^\ast)$  obtains. 
So now I can ask
A precise question : Is there a converse to the Kaplanski-Erdős condition i.e. if a $K$- vector space $V$ (automatically infinite dimensional !)  satisfies $dim_K (V) = card  (V)$ , is it the dual of some other vector space : $V \simeq E^\ast$? For example, is
 $\mathbb R ^{(\mathbb R)}$ - which satisfies the Kaplanski-Erdős condition ( cf. the "useful formula" below) - a dual?
A vague request : Could you please give "concrete" examples of duals and non-duals among infinite-dimensional vector spaces?
A useful formula : In this context we have the pleasant formula for the cardinality of an infinite-dimensional $K$-vector space $V$ ( for  which  you can find a proof by another of our friends, Todd Trimble, here )
$$ card \: V= (card \: K) \; . (dim_K V)  $$                 
 A: Let $K$ be a field of some size $\kappa$ and let $V$ be a $K$-vector space of dimension $\lambda$.  If $\kappa\leq\lambda$, then the dimension of the dual of $V$ is simply $2^\lambda$.  This is implies that vector spaces of large dimensions are duals iff
their dimension (and hence their size) is a power of 2.  I don't have the time right now to elaborate what happens if the dimension of $V$ is small relatively to the size of $K$.
A: The $\mathbf{R}$-vector space $\mathbf{R}^{(\mathbf{R})}$ has dimension $\operatorname{card} \mathbf{R}$ by definition, so it is isomorphic to $\mathbf{R}^{\mathbf{N}}$ (by the Erdős-Kaplansky theorem and because $\operatorname{card}(\mathbf{R}^{\mathbf{N}}) = \operatorname{card} \mathbf{R}$). So $\mathbf{R}^{(\mathbf{R})}$ is isomorphic to the dual of $\mathbf{R}[X]$.
In general, your precise question is equivalent to the following purely set-theoretical question (which seems difficult). By the useful formula, the identity $\operatorname{card} V = \operatorname{dim}_K V$ is equivalent to $\operatorname{card} K \leq \operatorname{dim}_K V$. Let $\kappa = \operatorname{card} K$ and $\lambda = \dim_K V$, and assume $\kappa \leq \lambda$. Does there always exist an infinite cardinal $\alpha$ such that $\kappa^\alpha = \lambda$? (here $\alpha$ is meant to be the dimension of the vector space whose dual is $V$). In general, Stephen's answer shows that there are counterexamples.
EDIT : in order to explain why the question is difficult, consider a field $K$ such that $\operatorname{card} K = \aleph_1$ (for example, one could take $K=\mathbf{Q}((T_i)_{i \in I})$ with $\operatorname{card} I =\aleph_1$). Take a $K$-vector space $V$ of dimension $\aleph_1$. Then $V$ is a dual if and only if there exists $\alpha \geq \aleph_0$ such that $\aleph_1 = (\aleph_1)^\alpha$, which amounts to say that $\aleph_1 = 2^{\aleph_0}$. In other words, $V$ is a dual if and only if the continuum hypothesis holds.
A: If $V$ is a vector space of infinite dimension over a field $K$, then the dimension of its dual is given by $\dim(V^*)=|K|^{\dim(V)}$. This is essentially just a restatement of what you have called the Kaplanski-Erdős theorem, because elements of $V^*$ correspond to functions $B\to K$, where $B$ is a basis of $V$, so $\dim(V^*)=|V^*|=|K^B|=|K|^{\dim(V)}$.
To answer your "precise question": Let $V$ be a vector space of dimension $\aleph_0$ over $\mathbb{Q}$. Then $|V|=\dim(V)$, yet $V$ is not isomorphic to the dual of any vector space, as its dimension is not of the form $|\mathbb{Q}|^\lambda$ for any infinite cardinal $\lambda$. Moreover, the same is true if $\aleph_0$ is replaced by any other strong limit cardinal, such as $\beth_\omega$.
Note that $\mathbb{R}^{(\mathbb{R})}$ is isomorphic to the dual of $\mathbb{R}^{(\mathbb{N})}$.
