My travels have been within the USA except for Toronto, Ontario, Canada. The extremes have been:
West to Texas,
South to Sarasota, Florida,
East and North to points in Maine.
Maximum distance from my current address: ~1400 miles to the NE (Maine).
You?
N - Point Barrow, Alaska
S - Santiago, Chile
E - Cyprus, Turkey
W - Apollo Bay (12 Disciples), Australia
Lived in Europe (Netherlands, France) and Singapore, traveled extensively over all continents except South America and Antarctica. Furthest north: Toronto (Canada), Copenhagen (DK), Fushun (China). Furthest south: New Zealand South Island (Invercargill), Cape Town (South Africa).
East and West becomes too relative if you travel this widely. 
Does being somewhere in an aeroplane count?
North: Somewhere over Alaska on a flight between Osaka and Detroit. (On land: Fort William, Scotland)
South: Southport, Tasmania
East: the 180th Meridian, many times, in a plane. (On land: Wellington, New Zealand)
West: the 180th Meridian, many times, in a plane. (On land: Honolulu airport, Hawaii – but in a plane) (On land, outside a plane: Victoria, British Columbia)
ETA: Maximum distance from current address: Fremantle, Western Australia, unless being on a ship in the Indian Ocean counts.
North: Inverness, Scotland.
South: Puerto Vallarta, Mexico
East: Budapest, Hungary
West: San Francisco, CA.
Yes, it should count in terms of actual position over points on the ground, but the idea I had was ground travel. The side issue of actually having both feet on the ground at whatever point seems like a way of limiting things for this thread’s sake.
List both, if you prefer, but stress the ground portion.
Thanks for pointing out the issue.
Excellent links! Thanks, Giles. We seem to meet often in these types of threads.
North: not quite to the Arctic Circle, on the highway north from Fairbanks, Alaska
South: Cabo San Lucas, Baja California
East: the island of Rhodes
West: Somewhere in the wilds of Alaska, west of Anchorage
Distance from current address: 6000 miles more or less to the east
North: Edinburgh, Scotland
South: Dunedin, New Zealand (I hadn’t realized this before, and it’s strangely appropriate, as Dunedin gets its name from Edinburgh.)
I can’t really say for East and West; I’ve gone all the way around the globe on one (very extended) trip.
East: Southern Maine
North: Southern Maine
West: San Francisco
South: Nassau
Strictly speaking from the Gaelic name of the Scottish capital, Dùn Èideann.
North–Canada, right above Glacier National Park
South-A couple hundred miles over the Gulf of Mexico
East-A couple hundred miles over the Atlantic Ocean
West-Pacific Ocean, from a beach in Washington State
North: somewhere in Scotland
South: Tasmania
East/West: Not sure how to define this. In terms of the 180th Meridian, Tasmania would also be the farthest east, and Lanai, Hawaii, the farthest west.
North - Amsterdam at 52 22 N
South - Gatun, Panama at 9 16 N
East - Hong Kong at 114 09 E
West - Muir Beach California at 122 34 West
These are places I’ve been on the ground. If flying counts, I passed approximately over the North Pole flying from Hong Kong to Newark, NJ.
Distance from current address: 8,047 miles over the pole to Hong Kong.
My coordinates came from Wikipedia. My distance came from here.
Never been off North America, and I’ve only traveled a couple miles from the border in Mexico. So:
North: Minnesota, around Minneapolis
South: Orlando, Florida
East: Baltimore, Maryland
West: Los Angeles, California
N: Banff, Canada
S: Puerto Vallarta, Mexico
E: New York City
W: Port Angeles, WA
I live in Philly, PA, USA metro area (NE USA).
North: Minneapolis, Minnesota
South: Southern Puerto Rico
East: Mumbai, India
West: Seattle, WA
.
Hm. Only counting places I left the airport:
North: A smaller city in Denmark, whose name I forget, which is a bit northwest of Copenhagen (55 40 N)
South: Fiji (18 S)
East: Fiji (175 E - can’t top that by much)
West: Portland, OR (122 39, to Seattle’s 122 20)
With airports, I can stretch further west to Honolulu and further south to Christchurch NZ (or wherever the big international airport is).
Correction:
North: Seattle, WA
South: Southern Puerto Rico
East: Mumbai, India
West: San Francisco, CA