In a very recent paper of MR Vaughan-Lee, it is proved that the number of $p$-groups of order $p^8$ and exponent $p$ is a polynomial (of fourth degree) in $p$.
Let us consider $p$-groups of order $<p^8$ and exponent $p$. I know that the number of $p$-groups of exponent $p$ and order $p$, $p^2$, $p^3$, and $p^4$ is (almost) a constant polynomial in $p$.
[By almost constant, I mean here, ignoring very few first values of prime $p$, the number of $p$-groups with required property is constant.]
Question: What about the number of $p$-groups of order $p^5,p^6,p^7$ and exponent $p$? Is it non-constant polynomial in $p$?
The number of groups of order $p^7$ and exponent $p$ is $7p + 174 + 2\gcd(p-1,3)$. See:
https://www.math.auckland.ac.nz/~obrien/research/p7/paper-p7.pdf
These were first computed by Wilkinson (who was my PhD student many years ago) here, but he made a couple of errors. It is stated in Wilkinson's paper that there are $34$ $p$-groups of exponent $p$ and order $p^6$ for $p \ge 7$. I would guess that it is also a constant for $p^5$.