4159^2 - 1 has 512 divisors. Gödel[7,3,1,1,1,1]
Bob Wainwright noticed that 5, 7, 29, 41 are associated with the largest dissected squares of increasing order. A good website on how to calculate Highly Composite Numbers is at http://www.eclipse.net/~dimona/juliannum.html. Juha Saukkola, David Terr, Warren Phillips, Stanislaw Jedrus, Shyam Sunder Gupta (http://www.shyamsundergupta.com/), Joseph DeVincentis, Oliver Lenhart, and Nathan Stohler all sent the answer.
68 divisors can be found with Gödel[6,4,1] = 2^6 * 3^4 * 5^1 = 25920. 25921 = 161^2. Claudio Baiocchi and others have pointed out this number isn't highly composite, since 4 smaller numbers have a larger number of divisors.