Number one, Socks the cat.
Although much has been said about Socks the cat in the intervening years since the Clintons left the White House - dangerous, brutal, one of the most effective political leaders of ours or any other time, it has been established definitively and totally that Socks elevated what a presidential pet could be. He turned the office from what had been once a merrily ceremonial position into one of the more powerful positions there is.
Number two, Nixon's Checkers the dog.
Checkers the dog is a myth, a legend in the White House. After that mysterious chandelier fell in the White House Grand Opera Ballroom, Checkers the dog had been reported throughout the White House skulking and terrified with his half mask.
Number three, the Kennedys' Pushinka the space dog
Little known fact. The Kennedy's were gifted a Russian dog that went to space. True fact. The dog who rarely ever barked would often spend hours looking off to the distance, as it may have seen something horrific in the grand, infamous, vastness of space.
Number four, Calvin Coolidge's pet hippopotamus Billy
Calvin Coolidge's pet pygmy hippopotamus, given as a gift to President Coolidge by the tire magnate Harvey Samuel Firestone, was affectionately named Billy or William Johnson Hippopotamus. Billy would often sit alone, playing with sticks, arranging them by twos, and then by threes, and then by fours, and then by fives, then disorganizing the arrangement and doing it once again. Boy, did they not love Billy, acknowledging that Billy was there as a hippopotamus simply living his life. Billy was small and quiet for a hippopotamus, so he was over a couple of hundred tons, but that was simply just Billy's life, until that one day. But that's a story for another day.