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The highest point in the Ethiopian Highlands is 14,928 feet above sea level. It is here that the Nile River begins its journey to the Mediterranean.

If the highlands were instead 21,737 feet above sea level--1500 feet taller than McKinley--would it have affected the Nile in any way?

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Cool! This is from my intuition (having been there), that it there would be no noticeable difference.

There are many sources of the water of the Nile, not just the region you're describing. However, a much higher plateau could mean additional rainwater, but reduced ice melting each spring.

(1) Lake Tana, the source of the Blue Nile "branch" is not the largest contributor to the volume or overall direction of the flow of the Nile.

(2) Mt McKinley is high, but also in an upper lattitude. Ethiopia has a perfect (IMHO) climate, because it is high up, but almost on the equator. Your weather will be much warmer than McKinley, although I don't know the climate-math.

(3) There would be many other effects (cultural, etc.), that I look forward to any of your follow-up questions.

PS- If you have a bucket list, please visit Lake Tana and all the little islands and Christian monasteries: easily one of the most amazing, little-known places on Earth; and it will help your story, maybe.

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