Without North America at all you can fly;

XONE4 Africa to Australia to South America to Europe to Africa or vv

XONE5 Africa to Asia to Australia to South America to Europe to Africa or vv

By flying to/from Santiago de Chile to Sydney or Aukland direct or to Papeetee and then surface to either OZ or NZ.

If you just want to avoid USA for visa or personal reasons yet still do a North America segment you can fly direct from Europe to

Canada, Mexico, Central America and Caribbean.

From Caribbean; you can do a bit of island hopping with BA and AA but will need to get to Havana to get to Cancun to get to Santiago.

San Juan Puerto Rico will get you to lots of islands (direct flights with Iberia from Spain) but not sure who stamps the passport!

From Canada; Vancouver has direct flights to Hong Kong with Cathay and Narita with Jal and also a flight to Mexico City with Jal as well.

Mexico City in itself is a dead end (as far as N/A segments go beyond YVR if US is out) and you would need to take surfaces if you don't fly to USA but does have flights to various South American cities.

You can fly from Europe to other Canadian cities but would need surface segments to Vancouver to leave the country as even the non USA destinations almost certainly stop in US.

Central America has direct flights from Europe but for hopping around looks very limited.

Most OW routes include USA because it makes life a lot easier when routing but with a bit of imagination 6 segments can be found in North America without US but bear in mind many of those segments could probably be purchased separately at very little extra cost on a point to point ticket.