The most popular position is that the individual should be allowed anything an infantry soldier would normally carry, which would include grenade launchers, squad automatic weapons, grenades and the like, but wait! There's more!
That pesky constitution includes in the part that describes the duties of the congress:
[Congress shall have Power...] To declare War, grant Letters of Marque and Reprisal, and make Rules concerning Captures on Land and Water;Letters of Marque and Reprisal are issued to private citizens to allow them to prosecute a war in support of the government. In order to do this with any credibility, the private citizen would need something resembling a warship, or at least a heavily armed merchantman. Since both land and water are cited, one supposes that a tank or artillery piece might also not be out of order. Want a tank? There's a place right down the road from me that sells them.
There is also a supply of used warships rusting in San Francisco Bay. Wouldn't an old destroyer make just the rig for trolling for Somali pirates? Sure you'd have to be Bill Gates to buy one, refurbish it and fuel it for a trip essentially around the world, but we can dream, no?