Cheap Business Class Fares to South America from $1,099
There is a big business class fare sale to South America going on right now according to this FT thread - and a lot of people are getting in on it, so if you haven't already, have a look at fares today to see if there are any that suit your needs. The main origins for these great fares seem to be Los Angeles and New York mostly, though if you find any others, please be sure to share them here.
You can find business class fares to various cities including Santiago, Chile (SCL), Buenos Aires (EZE), Lima, Peru (LIM) and Bogota, Colombia (BOG) are ranging in price from $1,099 to about $1,700 - all in! While that's still not quite mileage run material, it is a deep discount on fares that can range up to $6,000 in some cases, so they're worth looking into.
The best deals can be found on the ITA Matrix (though you can't book them directly there, you have to go either directly to an airline site or an OTA like Expedia or Travelocity), Hipmunk and Kayak. Sample schedules between March 1 and June 22 were used for this research; by the last week in June, roundtrip fares to all cities rise to $2,000+. All prices shown here are inclusive of taxes and fees. Here are a few sample South American business-class itineraries I found on various sites. As usual, I would recommend searching the ITA Matrix using its calendar search function and then honing in on your preferred routes, dates and airlines from there.
ITA Matrix
New York > Lima
$1,099
Avianca (JFK-BOG-GYE-BOG-LIM-BOG-JFK)
June 1-7
Los Angeles > Lima
$1,299
United Airlines (LAX-IAH-LIM)
New York > Lima
$1,307-$1,410
United (JFK-IAD-EWR-LIM-IAH-IAD-JFK)
March 1-June 14, various dates

Los Angeles > Buenos Aires
$1,602
Aeromexico (LAX-MEX-EZE)

Los Angeles > Santiago
$1,672
LAN Airlines (LAX-LIM-SCL)
New York > Santiago
$1,672
LAN Airlines/TAM (JFK-SCL-GRU-JFK)
April 3-May 16, various dates
New York > Buenos Aires
$1,699
LAN Airlines (JFK-SCL-EZE)
April 13-19, May 1-7, May 11-17, June 15-22
Kayak
New York > Bogota
$1,247
American Airlines (JFK-MIA-BOG)

Los Angeles > Bogota
$1,386
Avianca (LAX-SAL-BOG)
New York > Lima
$1,368
American Airlines (JFK-DFW-LIM)
April 13-17

New York > Santiago
$1,672
LAN Airlines (JFK-SCL - Nonstop)

New York > Lima
$1,389
American Airlines (JFK-DFW-LIM-MIA-JFK)
May 4-9
Hipmunk
Los Angeles > Bogota
$1,485
United Airlines (LAX-IAH-BOG)

Los Angeles > Bogota
$1,485
Copa Airlines (LAX-PTY-BOG)
While some of these aren't exactly mileage-run worthy, they are still deeply discounted itineraries and the routing on many of them is easy and at decent times. For instance, this itinerary from JFK to Buenos Aires is on LAN via Santiago and you would get to fly LAN's new 787.

This is booking into I class, which earns 100% miles on American, so for $1,699, you're earning 10,564 miles roundtrip - about 16 cents per miles, so definitely not a mileage run, but definitely deeply discounted business class on a nice product.
These were just some quick examples I found, but if you find any more, please be sure to share them in the comments below!
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