This is quite fun actually. I live in Milan (Italy) and my cousin lives in Aukland (NZ). These cities are listed as being the furthest from each other at more than 18,000 km.
Incidentally as a kid I was quite obsessed with the idea of visiting the antipode of my home's exact geographic location, but sadly, when I finally got around to calculating it, I discovered it lay in a nondescript expanse of ocean.