Instead of limiting entries per person (check the numbers over the past few years and you'll see the average # is about 4 per), or making a whole new round of judging (there aren't the resources for this) how about limiting the number of entries per category.
There are only 750 entries for 28 categories, cap the entries per style category at 40. Total still capped at 750 (no not every category would be able to reach it's max, but most don't hit 40 anyways)
Want to enter something in 10, 13, 14, 16, 18, 23? Then you better enter the first day as soon as it opens, because they will fill up. Have entries in the less overloaded styles? There will be about 500 entries left per region for those once the Big 6 categories fill up. Don't want to risk not getting your Melomel or Doppelbock in? Go ahead and enter the first day as well, but you'll probably be able to wait
Encourages people to brew new styles, go out of their comfort zones instead of Pale Ale, IPA, Stout, Belgian, Pale Ale, IPA, Stout, Belgian ... These make up about 40% of NHC entries!
Simplifies Judging, no need to setup 4, 5, 6 judging tables to pass beers to a final round in the overloaded categories