Neither the numbers 2^33 and 5^33 contain any zero. This is quite odd considering the product of 2 and 5, the base is ten, and the number of digits used; 8589934592 and 116415321826934814453125, respectively.
33 is probably the highest number where this is true. AFAIK, no one has proved this to be true, so I empiracally tested all values up to 10000 and failed to find one. Under 33, however, there are 9 results: 1,2,3,4,5,6,7,9, and 18.
FWIW, the numbers at 10000 have 3011 and 6990 digits in them.