If n is even, you can go infinitely high.

Consider n = 2m and the product of (10^n - 1) and (10^n - 10^m - 1)

Since both of these are smaller than 10^n they each have at most n digits. 10^m - 1 is necessarily smaller than 10^n-1 and so the first digit of the second number is 9, and each number has exactly n digits. More descriptively, the first product is n 9's while the second is m 9's followed by m-1 0's followed by a 1.

The product (using 2m = n to eliminate n) is:

10^4m -10^3m + 10^m - 1 = 10^3m(10^m - 1) + (10^m - 1)

Now, 10^m - 1 is a sequence of m 9's, and the 10^3m factor shifts that pattern 3m digits to the left so there can be no overlap. The resulting number then has m 9's followed by 2m 0's followed by m 9's. Such a number is palindromic.