I've seen this used to take either Basic Weapon Training (Universal), Flame Weapon Training (Universal), Melee Weapon Training (Universal), or Pistol Weapon Training (Universal). I've even seen Heavy Weapon Training (Bolt) for a character that wanted to take a Heavy Bolter as his starting Acquisition. Another character asked if he could take Exotic Weapon Training (Shuriken Catapult) along with Hatred (Eldar) from the Hand of War along with a good-craftsmanship Shuriken Catapult as his starting Acquisition. This seems reasonable, but it does leave me wondering if there is any real limits on what Weapon Training Talents would be allowable. Other than Xenos Weapon Training (Ork) which specifies that it is only applicable to Orks, are there any hard limits to what The Hand of War may provide?

Nope, it's entirely open to whatever Training talents that the player wants, and can justify to the GM so that they allow the player to have them.

Remember, starting RT characters represent exceptional and unique individuals in their prime. They already have had a lifetimes worth of experience (from a modern day point of view) at the start of the game, as described by their origin path. They may well have travelled and trained extensively.

So its not that unreasonable that they would have the chance to get trained in just about anything. The PF aquisition is the only real limit here.