If doing the 7 Summits is only about saying you have done them, 7 will probably do.
If you are climbing the peaks for the mountain experience, the people you meet and the cultures you pass through, then you may as well do 9 or 10 of the 7 summits. On some lists you will find up to 11.
Here are the ones that are undisputed and you should climb, and then there are a few more to add to your list…

Everest – undisputed, no matter what your geology, political or tectonic plate beliefs are, you’ll just have to do it. Even if you have to wait in line.
North Side and the North Ridge shown here, perhaps a touch harder, but you don’t have to go through the Khumbu Icefall. Both sides laden with history, from Mallory and Irvine on the North, to Hillary and Tenzing on the South side. If you are looking for obscurity, danger and a real adventure, head over to the Kangshung or East Face.

Aconcagua – undisputed, until they build a big tower on Ojos-del-Salado, which has on occasion been claimed as taller. On current GPS calculations they are just 68 meters apart. Choose the Polish Glacier or South Face and you could well find yourself all alone.

Denali – undisputed, unless you feel the land bridge from South America to North America makes it all one continent, then maybe you should climb Aconcagua twice?
Denali also stands out as by far the highest of the 50 states for Highpointers, those ascending the highest point in every U.S. state.
Taking the West Buttress in season you will have plenty of company, expect a social climb and lots of hopefully like minded climbers. If you want a wilderness experience, you can go just about anywhere else on the mountain and you will likely be all on your own.

Kilimanjaro – undisputed. Though if you want to see the ‘snows of Kilimanjaro’ you need to go soon.
The classic “Coca-Cola” route is the most heavily trafficked, anything else you will find far less company.

Elbrus – now it gets tricky. You can draw the European line at the Caucasus, which separates Elbrus out from Asia, or further east, and you have Mt Blanc.

Best to climb both peaks, as Mt Blanc is such a great climb it is not to be missed.

Carstenz and Kosciusko – geographically and land mass wise, Kosciusko wins. Carstenz is certainly a great climb, much more fun as a climbing adventure – again, best to do both. And if your continental definition extends to ‘Oceania’ you certainly have to climb it.
There are so many new route possibilities on Carstenz – it is a real mecca to be explored – just look at it – the entire face has two known routes.

Recent articles have advocated for a New Zealand continent, which means you can add Mt Cook to your list, certainly a worthy objective, with its fierce weather, steep slopes and a classic Ed Hillary route or two – rarely repeated and a good reminder why he and Tenzing made the top first.
And you can go for the 2nd 7, the second tallest peak on all continents, certainly more challenging, then the 3rd 7 summits, which has so far only been done once.
Perhaps more interesting are the 7 volcanoes, which gets one out to Iran to do Damavind, down to Antarctica on Sidley and over to the previously mentioned Ojos-del-Salado.
And if you have done the 7 summits, you already have Elbrus and Kilimanjaro, so only 5 to go, some great new countries to visit and you can get back to Papua for the obscure Mount Giluwe. And there are also the rarely repeated climbs, like the SW side of Vinson, relatively easy to access and with only 2 unrepeated routes, 003 and the Rolex Ridge.
