Take $n$ non-negative dependent random variables $X_1,...,X_n$ with $Pr(X_i \leq t) = t, t\in[0,1]$ for every $i$ (uniform marginal distributions).
What is an example of a joint pdf for $X_1,...,X_n$ (with the given common marginal distribution), such that $E[\min_i X_i] = 1/2$?
Assume that $(X_1,X_2,\ldots,X_n)$ is a solution and let $M=\min\{X_1,X_2,\ldots,X_n\}$.
Then, for every $i$, $M\leqslant X_i$ almost surely and $\mathbb E(M)=\frac12=\mathbb E(X_i)$ hence $M=X_i$ almost surely. This holds for every $i$ hence $X_1=X_2=\cdots=X_n$ almost surely.
In which case, naturally, $\mathbb E(M)=\frac12$ indeed holds, and $(X_1,X_2,\ldots,X_n)$ has no joint PDF.
On the other hand, for every positive $\epsilon$, there exists $(X_1^{(\epsilon)},X_2^{(\epsilon)},\ldots,X_n^{(\epsilon)})$ with uniform marginals and with a joint PDF, such that $\mathbb E(M^{(\epsilon)})\geqslant\frac12-\epsilon$.