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Gilles Villeneuve vs. Didier Pironi
The tragedy of this rivalry, not to mention the controversy, make it a heated topic of debate. Villeneuve was enraged after Pironi defied a team order to hold station in the 1982 San Marino Grand Prix, and vowed never to talk to him again.
He didn't, because Villeneuve crashed fatally at the next race at Zolder trying to beat Pironi's time in qualifying. A brief but heated rivalry but, given that it spanned two Grands Prix, it seems unjust to put it in the top five.
Alain Prost vs. Nigel Mansell
Nigel Mansell had quickly established himself as a fan favourite at Ferrari, but new arrival Alain Prost made for a strained relationship in 1990.
The Brit suspected Prost was getting better equipment and even announced his retirement halfway through the year. Mansell blocked Prost at the start of the Portuguese round, severely compromising his race.
Though he remained in the sport after changing his mind, he did quit after winning the title in 1992, because Prost would have been his Williams teammate in 1993.
Mika Hakkinen vs. Michael Schumacher
Naturally, these two come close to cracking any list. But Schumacher's Ferrari was not good enough to give the Finn a proper challenge in 1998, and his broken leg at Silverstone denied a '99 battle. 2000 was the peak of this tussle, with Schumacher emerging on top to deliver his first world title for Ferrari.
Fernando Alonso vs. Lewis Hamilton
The sole season Alonso and Hamilton spent together at McLaren saw sparks fly on a number of occasions.
Alonso, the double world champion, was being beaten by his rookie teammate. At Hungary, he attempted to blackmail team boss Ron Dennis—this was going on amid the Spygate scandal in which McLaren had obtained confidential data from Ferrari.
Both drivers attempted to baulk the other's qualifying efforts. From there, the relationship was in disrepair. It cost the pair the title, as their desire to beat one another allowed mistakes to creep in and Ferrari’s Kimi Raikkonen to claim the crown.
Jim Clark vs. Graham Hill
Two great British drivers (no pun intended) who demonstrated perfectly that hatred/dislike/controversy need not fuel an intense rivalry.
The pair had a more-than-amiable relationship off-track, but a fierce competitiveness on it. They shared several titles between them in the 1960s, including two years as Lotus teammates in which they were both champions (Clark '67, Hill '68).