Batteries should be AA NiMH rechargeable. Using any other battery chemistry type is definitely not good and can be dangerous. Although this sort of activity does not void the warranty, it is not recommended.
There are a lot of different types of batteries out there with different brands, ratings, and claims—far too many for Magic-Flight to test each one. It is also true that the Launch Box uses the batteries in a somewhat unusual manner—however, not so different that most batteries of the correct classification should work reasonably well enough. The batteries to get are of the NiMh type, and need to be able to store at least 2400 mAh. AAA batteries are too thin to work—the Launch Box is made specifically to fit AA batteries. There is some variation between battery makers, so some fit better than others.
The included batteries are sure to provide enough power for one full trench—you might get more than that, but that is a reasonable expectation. Batteries that have a higher mAh rating could last for more than one full trench (twelve draws, 15 seconds each draw, 10 to 20 cc per sec, vapor just visible in the draw channel but not otherwise at least 55% of the time). The device is made as nearly perfectly energy efficient as the physics and materials allow, so it is unlikely that the "one battery per trench" situation will change anytime soon.
Surprisingly, Magic-Flight has found that having the battery be stronger than that needed for one trench actually has two disadvantages: 1) users are constantly wondering if the battery is okay on the second and third trench, and 2) if a user happens to accidentally leave the unit "on", stronger batteries tend to get hotter (in proportion to their storage capacity). Magic-Flight therefore strongly suggest that people only get other batteries for the Launch Box after they have developed a good set of usage habits with the provided batteries and are familiar with the capabilities of the device.