Comparison of means: Unknown distribution and Unequal Variances

I am doing an experiment, on certain random numbers with an extreme
value kind of a distribution, when the true underlying distribution is
unknown. The problem is something like this:

Interested in knowing the differences in means of min{X1,.....,Xn}
where Xi ~ iid unknown dist, for different values of n.

Obviously many assumptions of ANOVA are violated. To complicate things
further Levene's test p-value is 0.000, so we are nowhere in the range
in which the F-test remains robust. Box-cox transformation is not
working. Since, the distribution of Xi's is unknown, I was thinking
about some non-parametric test. But I am pretty sure that permutation
tests and there special cases (the Wilcoxon's tests) won't be valid
because of the high levels of variability. Does anyone has any
suggestions?

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