[mturmon]

You have given a nice clean answer that does not make any errors (such as talking about the likelihood as a density in parameters, which of course it is not). Thanks for writing it down.

The only other thing worth adding to what you have written is that the likelihood is a product of N factors.

As such, it will essentially always diverge toward infinity (if the density factors are on average greater than 1) or collapse fast towards zero (if the factors are on average less than 1, as in your example and in OP).

So this very structure (arising from the IID observations) implies that no “stable” density will pop out. It’ll always blow up or down!

One way to stabilize things is to take (1/N) times the log of the likelihood. Then you will indeed converge to something familiar - the entropy, - E log p(x).