Pull entire row based on MAX(Column) MySQL
If you don't care about ties and want one row only and always one row, then the following could be used. With an index on (Score), should be as efficient as possible:

SELECT ID, Name, Score
FROM TestTable
ORDER BY Score DESC 
LIMIT 1 ;


If you care about ties but still want only one row, you'll have to decide how ties should be resolved and modify the ORDER BY accordingly. If, for example, the row with the first name in lexicographic order is wanted in case of ties, then use the following - and an index on (Score, Name):

ORDER BY Score DESC, Name ASC


If you want all tied results, then the query by @Dan in the other answer should be the best. The index on (Score) would be sufficient for that.