Melhor

tl; dr: 5 vezes

Harry menciona em vários locais que este é "o melhor dia", vou descrever esses exemplos a seguir. Tenha em mente que "o melhor de todos" é um termo comum que os jovens usam quando se sentem exultantes (o mesmo vale para o inverso "o pior de todos"); nós gostamos de exagerar.

Harry Potter e a Pedra Filosofal
Os Dursleys e Harry visitam o Zoológico de Surrey para comemorar o 11º aniversário de Dudley. Harry parecia apreciar a visita ao zoológico e sentiu que era um dos melhores dias que ele teve em um tempo:

[the lemon ice pop] wasn’t bad, either, Harry thought, licking it as they watched a gorilla scratching its head who looked remarkably like Dudley, except that it wasn’t blond.
Harry had the best morning he’d had in a long time. He was careful to walk a little way apart from the Dursleys so that Dudley and Piers, who were starting to get bored with the animals by lunchtime, wouldn’t fall back on their favorite hobby of hitting him...
^{-Harry Potter and The Philosopher's Stone, Chapter Two (The Vanishing Glass).}

Harry Potter e a Pedra Filosofal
Harry descobre que ele é um bruxo em seu aniversário de 11 anos. Ele sente que teve o melhor aniversário de todos os tempos:

“You all right, Harry? Yer very quiet,” said Hagrid.
Harry wasn’t sure he could explain. He’d just had the best birthday of his life — and yet — he chewed his hamburger, trying to find the words. ...
^{-Harry Potter and The Philosopher's Stone, Chapter Five (Diagon Alley).}

Harry Potter e a Pedra Filosofal
O primeiro Natal de Hogwarts de Harry, e acabou sendo o melhor Natal até agora:

It had been Harry’s best Christmas day ever. ...
^{-Harry Potter and The Philosopher's Stone, Chapter Twelve (the Mirror of Erised).}

Harry Potter e a Pedra Filosofal
A primeira festa de fim de ano de Harry. Depois de derrotar Quirrelmort, Harry comparece à Festa do Fim do Ano, na qual o Prof. Dumbledore anuncia a Grifinória como a vencedora da Copa:

It seemed as though life would be back to normal next year, or as normal as it ever was at Hogwarts.
It was the best evening of Harry’s life, better than winning at Quidditch, or Christmas, or knocking out mountain trolls . . . he would never, ever forget tonight. ...
^{-Harry Potter and The Philosopher's Stone, Chapter Seventeen (The Man with Two Faces).}

Harry Potter e o Pretor de Azkaban
Harry e sua equipe treinam para a partida contra a Corvinal naquele mês e durante uma das práticas, Harry diz que foi a melhor prática de Quadribol de todos os tempos:

It was the best practice ever; the team, inspired by the presence of the Firebolt in their midst, performed their best moves faultlessly ...
^{-Harry Potter and The Prisoner of Azkaban, Chapter Thirteen (Gryffindor versus Ravenclaw).}

Pior

tl; dr: 4 vezes

Harry Potter e a Pedra Filosofal

It was lucky that Harry had tea with Hagrid to look forward to, because the Potions lesson turned out to be the worst thing that had happened to him so far.
^{-Harry Potter and the Philosopher's Stone, Chapter Eight (The Potions Master).}

Harry Potter e a Câmara Secreta
Curiosamente, o primeiro capítulo é chamado de "The Worst Birthday"; mas não há menção de Harry sobre o pior de tudo.

Harry Potter e a Câmara Secreta
Quando Gina foi sequestrada pelo monstro na Câmara Secreta:

“The Heir of Slytherin,” said Professor McGonagall, who was very white, “left another message. Right underneath the first one.
‘Her skeleton will lie in the Chamber forever.’ ”
Professor Flitwick burst into tears.
“Who is it?” said Madam Hooch, who had sunk, weak-kneed, into a chair. “Which student?”
“Ginny Weasley,” said Professor McGonagall.
Harry felt Ron slide silently down onto the wardrobe floor beside him.
“We shall have to send all the students home tomorrow,” said Professor McGonagall. “This is the end of Hogwarts. Dumbledore always said . . .”
...
It was probably the worst day of Harry’s entire life. He, Ron, Fred, and George sat together in a corner of the Gryffindor common room, unable to say anything to each other. Percy wasn’t there. He had gone to send an owl to Mr. and Mrs. Weasley, then shut himself up in his dormitory.
^{-Harry Potter and the Chamber of Secrets, Chapter Sixteen (The Chamber of Secrets).}

Harry Potter e o Cálice de Fogo
Quando Ron parou de falar com ele porque todos achavam que ele havia colocado seu nome NO CÁLICE DA FYAH!

The next few days were some of Harry’s worst at Hogwarts. The closest he had ever come to feeling like this had been during those months, in his second year, when a large part of the school had suspected him of attacking his fellow students. But Ron had been on his side then. He thought he could have coped with the rest of the school’s behavior if he could just have had Ron back as a friend, but he wasn’t going to try and persuade Ron to talk to him if Ron didn’t want to. Nevertheless, it was lonely with dislike pouring in on him from all sides.
^{-Harry Potter and the Goblet of Fire, Chapter Sixteen (The Chamber of Secrets).}