Was gonna suggest cg125, totally reliable for me despite my bulk and natural urge to rag the backside off everything.
Good on fuel, cheap bits, it's a honda, so nothing falls off...
That said I sold it to a guy at work who bought it to learn on and he fannied around with the carb settings for some reason and knackered it. So they can be broken if you're an idiot. D'oh.
Round the world? Well the adventure bike is the obvious choice so... BMW K1600 or Goldwing. Fairing enough to deflect storms, grunt enough to gobble miles, hills etc. Good on fuel, enough space in the boxes to pack a machine gun and ammo
Just the advice to bear in mind - "Stick to the roads. Stay off the moors."