Is there any real Algorithm with a time complexity O(n^n), that isn't just a gimmick?

I can create such an Algorithm, like computing n^n in O(n^n) / Θ(n^n):

long n_to_the_power_of_m(int n, int m) {
    if(m == 0) return 1;
    long sum = 0;
    for(int i = 0; i < n; ++i)
        sum += n_to_the_power_of_m(n, m-1);
    return sum;
}

(needs more than 4 minutes to compute 10^10)

Or other way around: Are there any Problems, which cannot be solved better than in O(n^n)?

algorithm complexity-theory big-o
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