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It would be extremely difficult. Some of the highpoints have only been climbed once and are quite difficult. For example, I know that Julianatop was first climbed in 2011 and I don't think it has been repeated.If you were resourceful enough, that country list is definitely finishable.
The Pacific countries have some hard ones as well. French Polynesia's summit used to be fairly reasonable (though exposed) until the forest fire stripped the slopes and burned the fixed ropes. A competent climber could climb it without the fixed ropes, but the slopes are extremely dangerous since the forest fire. It would be somewhat suicidal now days.
I know Pico Cão Grande has been and can be climbed, but it does look challenging:
Maybe it's not so bad though as there are trip reports on it.
Maybe some of the country highpoints are still unclimbed? I'm not sure on this though. I know some of the island highpoints (even on tourist islands such as Bora Bora) are still virgin.
Some country highpoints are definitely off limits due to military installations on their summits (i.e. Oman, Saudi Arabia, Yemen, and Sri Lanka), while some others are littered with landmines. I wouldn't want to even attempt peaks such Noshaq right now, though the climb itself is straight forward (at least for a 24,000+ foot peak). Peaks such as Emi Koussi are pretty risky right now, though I want to do that one someday. Of course K2 is also a country highpoint as is Kanchenjunga.
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Thoughts?
I wonder how many highpoints do we have between us 14ers.com members?
Although I'm not that much of a highpointer, I have the following:
Aruba
Australia
Guatemala
Costa Rica
Curacao
Dominica
Dominican Republic
Ecuador
El Salvador
Grenada
Guatemala
Guyana
Honduras
Hong Kong (the state department still classifies this as a country, but I don't know why??)
Jamaica
Saint Lucia
Saint Vincent and the Grenadines
Singapore
Spain
Thailand
Trinidad and Tobago
Additional failed attempts are as follows:
Chile (high winds)
Russia (very bad weather in the off season)
Mexico (sick with flu and hut/campsites too crowded)
I know a lot of other members have done Denali. I think some have done Logan. For sure, 14ers.com members have France, Italy, Peru, Tanzania, Nepal, Pakistan, and Argentina. I'm sure that there are many others too.