Despres grew up with parents who ran restaurants. He sold approximately 6,000 bottles of wine to fund his debut at the 2000 Dakar Rally on a Honda. He had already won the French Enduro Championship in 1998, his debut year in that discipline. His first Dakar start resulted in sixteenth place; the following year on a BMW he finished thirteenth and took his first stage victory.
After switching to KTM from 2002, Despres steadily progressed through the field: DNF in 2002, second in 2003, third in 2004. He took his first Dakar victory in 2005, defeating Marc Coma. Over the following decade he and Coma would each win five Dakar titles, their rivalry defining the motorcycle category across that era. Despres's subsequent wins came in 2007, 2010, 2012, and 2013, all on KTM.
His 2012 victory was his fourth, equalling the all-time motorcycle record held by Edi Orioli. That edition featured a controversial incident on 10 January when Portuguese rider Paulo Gonçalves pulled Despres free from a mud trap; Despres did not assist other riders who became stuck shortly afterwards. Race organisers modified the course, credited Despres with lost time, and imposed a six-hour penalty on Gonçalves for accepting outside assistance. Despres reclaimed the overall lead on Stage 13 of the final stages after Marc Coma suffered a mechanical failure.
In 2013 Despres won for a fifth time, making him one of the most successful Dakar motorcycle competitors in the event's history, second on the all-time list behind Stéphane Peterhansel's twelve combined victories. Between his motorcycle Dakar campaigns Despres also held the FIM World Championship in 2003 and 2009.
Alongside Dakar, Despres built an extensive record across other major rally-raid and extreme enduro events. He won the Red Bull Romaniacs in 2004, 2005, and 2007, and took consecutive Erzberg Rodeo wins in 2002 and 2003. His rally-raid wins included multiple victories at the Rallye du Maroc (2000, 2003, 2010, 2012), the Tunisia Rally (2004, 2005, 2009), and the Rally dos Sertões in Brazil. He also won the Orient Rally on multiple occasions.
Despres moved to the car category in 2015, joining Peugeot's factory squad alongside Carlos Sainz and Stéphane Peterhansel — who had himself made a successful motorcycle-to-car transition. Despres and co-driver Gilles Picard finished thirty-fourth in their debut at Dakar 2015. In 2016, with David Castera as co-driver in a Peugeot 2008 DKR, he improved to seventh at Dakar, and the pair won the Silk Way Rally outright. In 2017 Despres and Castera repeated the Silk Way Rally win in a Peugeot 3008 DKR and claimed third overall at Dakar in the car category — his best car Dakar result to that point and his first car podium at that event. In 2018 he was a fully contracted Peugeot Total factory driver.
Post-Peugeot, Despres continued in cars: he ran a Mini X-Raid in 2019 Dakar with Jean-Paul Cottret as co-driver, finishing fifth, before an SSV class entry in 2020 ended with retirement.
Despres's five motorcycle Dakar victories place him among a very small number of drivers to have reached that total at the event. His protracted duel with Marc Coma — five titles each between 2005 and 2015 — produced what is widely considered the definitive rivalry of the Dakar's modern era in the motorcycle category.
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