I will slightly expand upon my comment.  For N a prime, one has the desired product being
(N-1)!.  For N=pq, a product of two primes, the desired product is N! * pq/[p!q^p q!p^q] .  I have not worked it out, but one should note a similarity between this representation and that of the cyclotomic polynomial Phi_pq(x).  For square free N=pqr I imagine an analogous relation holds.

If you want something rougher but still in the ball park, consider (N!)^2 as product of terms of the form (n+1-i)i, which for many i look like o(N^2).  Then the desired product
is something like phi(n)/n fraction of those terms, perhaps with a power of e to take
care of the discrepancy.  So I nominate without proof [((N!)^(1/N))/e]^phi(N) as a rough
estimate.  I imagine provable upper and lower bounds are a mere exponential factor away from this
estimate.

Gerhard "Ask Me About Rough Estimates" Paseman, 2012.05.18