(at least) 4 extra days in Argentina (with or without me)

Apart from Buenos Aires, I suggest you taking at least 4 extra days to get to know other cities.
Explore the first couple of options if you want to keep your Budget low.
Another option is to take a couple of day trips not far from Buenos Aires.
Days out:
Cordoba is one of the largest cities of the country and it has nice mountain villages around it. I could drive you there and spend those days with you.

• First Option: CORDOBA (With me, by car)

1) 7hs drive to Cordoba City
2) Córdoba hills villages: Villa General Belgrano/ La Cumbrecita:
http://www.villageneralbelgrano.com/fotos/index.asp?scc=26
http://www.villageneralbelgrano.com/fotos/index.asp?scc=28
http://www.argentinaturistica.com/2vgbiresenia.htm
3) Córdoba City
4) Return to Buenos Aires (we could stop by Rosario on our way back: map )
The following is probably the easiest option: I think I mentioned my family owns an apt in Pinamar just in front of the beach. If you are going to visit Brazilian beaches I´m not sure it´s worth it, but here you have a good link: http://www.argentina.ar/_es/turismo/guia-del-viajero/C3013-pinamar-entre-las-dunas-y-el-bosque.php Horse-riding at sunset is a must, and the town is really pintoresque.

• Second Option: PINAMAR (With me, by car)

1) 4hs drive to Pinamar
2) Pinamar + 15-minute-trip to Cariló (Pictures; more pics )
3) Pinamar
4) Return to Buenos Aires (maybe we could stop by La Plata city on our way back: map)
Another choice is to take a number of day trips from Buenos Aires. Short distances but worth-watching places.

• Third Option: SHORT TRIPS (With me, by car)

1) Day trip to Colonia, Uruguay (by ferry: the prices are in pesos )
2) Day trip to Rosario (drive map)
3) Day trip to La Plata ( drive map)
4) Day trip to Tigre: 45 minutes ride from Buenos Aires

The last two options are probably the best, but it requires airfares.
Anyway, you should take a look at the Patagonia part, to see how nice it is. It is much more expensive, but totally worth it. http://www.en.argentina.ar/_en/tourism/patagonia/index.php
Last summer I went to The Road of the 7 Lakes and Bariloche. You´ve seen the pictures, it is incredibly beautiful, but probably very European-like. Glaciers and El Calafate will be surely something very different. You can even take a trekking tour on the Perito Moreno glacier.

• Fourth Option : PATAGONIA (By yourself, by plane) 

1) Trip to Patagonia region (by plane): Bariloche + The Road of the 7 Lakes (day excursion map; Pictures, and more pics) / Or / Glaciers and El Calafate (map)
2) Patagonia
3) Patagonia
4) Return to BA
As a fifth option (intermmediate in terms of budgets and one of the best in terms of points of interest), I´m including a trip to Salta, in the North region. It might be hot, but there´s a unique Baroque style in the area that is worth seeing it. There are good wineries in the area and the landscape is beautiful. You will get to see native people, with their typical costumes in their everyday life.

• Fifth Option : SALTA (By yourself, by plane )

1) Trip to Salta (by plane):
2) Salta: Day trips to Calchaqui Valley or Cafayate wineries
3) Day trips to Calchaqui Valley or Cafayate wineries
4) Salta city Return to BA
All the other regions are also great. Cuyo and Litoral, but you must know Summer is mostly too hot to go there.
Anyway, I´m including those regions here:

• Sixth Option: IGUAZU FALLS (By yourself, by plane)

You can take a plane to Iguazu Falls (in the border with Brazil)
http://www.lan.com/en_us/sitio_personas/destinos/argentina/iguazu.html

• Seventh Option: MENDOZA or SAN JUAN (By yourself, by plane, or maybe we can drive there, but we´ll need a few extra days :)

You can also fly to Mendoza or San Juan. Here it´s hot but since they have a very dry weather, it´s easier to cope with it. In February they have the Vendimia (Harvest) festival in the wineries.
Mendoza: http://www.turismo.mendoza.gov.ar
San Juan: http://www.travelsanjuan.com.ar/ingles/home.html

http://www.thegrapevine-magazine.com/features_wine_country/features_wineharvest.htm