Are there $n$ groups of order $n$ for some $n>1$?

Unfortunately, we have a complete picture only up to around order of 2100. For the rest, we can calculate some, but these are either fixed and small, respectively smaller than 2100, or comparatively smaller than prime factors, or they are evidently much larger than $n$.

Still, we can conjecture. And the conjecture says NO.

I have created two graphs. Both related to something I will call distance $D(n)=|N(n)-n|$.

First direct graph, $D(n)$ is displaying the distance based on order. As you can see it looks linear, because most of the groups have small and fixed order. Having anything else than small order is kind of exception.

The second graph is giving the overall number of distances less than a given number. Basically it says in how many cases (for how many group orders) the distance is less than a given number. Again it is strongly linear, first few values 2,3,6,7... , meaning there are 2 group orders whose distance is 0 (it is order 1 and 0), 3 group orders with distance 1 or less, 6 with 2 or less, 7 with 3 or less. A strong linear shape is suggesting that small orders are very much fixed and it is very unlikely that 0 will ever suddenly appear as orders grow.

For both of the above I would strongly say no: there is only one possible outcome $1=N(1)$

The heuristic reason is that the number of groups depends mainly on the factorization of $n$. For all larger numbers this is either fixed into a small constant or depends on one or two smaller factors. As soon as we have several factors involved, the number of possible groups is going to jump over the order easily.

If there is an exception to this, this group should have had some special theoretic properties we are currently not aware of.