2006 AMC 12A Problem 4

The two minutes digits sum to at most $5 + 9 = 14,$ at $59$ minutes past the hour. For the hour, a single digit $9$ gives digit sum $9,$ which beats any two-digit hour $(10, 11, 12$ give at most $1 + 2 = 3).$ The largest total is $14 + 9 = 23,$ occurring at $9\!:\!59.$ Thus, the correct answer is E.