The most common and most likely answer to this crossword clue is the 10 letter word SUPERPOWER
I've seen this crossword clue in The Daily Telegraph
S
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More clues leading to the result SUPERPOWER
Buddy explains!

The straight part
We know that "dominant country" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dominant country = superpower.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SUPERPOWER mean?
SUPERPOWER noun- Excessive or superior power.
- A sovereign state with dominant status on the globe and a very advanced military, especially the Soviet Union or United States.
- A fictional extraordinary physical or mental ability, especially possessed by a superhero or supervillain.
- Electricity generated in a large plant that is tied into a regional network, on a larger scale than was common in the early years of commercial electricity production.
- (mathematics) A tetration (also known as a power tower and the inverse of a superlogarithm).