This paper by Alaoglu and Erdős proves that if $p$ is the largest prime divisor of a highly composite number (HCN) $n$ then the exponent $k_q$ on another prime $q$ dividing
$n$ satisfies
$$\log\left(1 + \frac{1}{k_q}\right) > \frac{\log q \log 2}{\log p} + O\left(\frac{(\log \log p)^3}{(\log p)^3}\right),$$
$$\log\left(1 + \frac{1}{k_q+1}\right) < \frac{\log q \log 2}{\log p} + O\left(\frac{(\log \log p)^3}{(\log p)^3}\right).$$
(See Theorem 12.)
For sufficiently large $q$ they claim that $k_q$ is either $\Lambda$, $\Lambda + 1$, or $\Lambda - 1$, where $$\Lambda = \left\lfloor \left(2^{\log q/\log p} - 1\right)^{-1} \right\rfloor.$$ (See Theorem 13.)
When $q = 2$ this latter formula appears to be close, although it is not exactly correct, as there are more than three possible values for the power of 2 that divides a highly composite number with largest prime factor $p$. See, for example, this list of the first 1200 HCN's, which includes their prime factorizations.
This, together with Derek Jennings' answer on the order of $p$, should give you the estimate of the highest power of 2 that divides $n$.