In pretty much every gen, Dragon Quest is more popular. In the NES gen, DQ destroyed FF, coming out 2 1/2 years before it. The first Final Fantasy game came out in Japan only 2 months before Dragon Quest III, which was one of the fastest selling games on the NES in Japan and went on to sell over 3.6 million (impressive for any generation).
Sales have grown for each title following, eventually topping out to Dragon Quest VII on the PSX, the best selling Japanese game in history with over 4.1 million units sold in just Japan. Even the almighty Final Fantasy VII didn't match that in Japan. And this trend of overwhelming sales continued into the PS2, with Dragon Quest VIII topping all the Final Fantasy titles.
Now it is not so easy to compare the two, as Dragon Quest has made a firm place on the Nintendo platforms and Wada has pretty much milked Final Fantasy out more than Sonic and Madden at Mardi Gras. But one thing is for sure. All the Dragon Quest titles still continue to sell extraordinarily well. Will all of the titles released so far having sold a million or better in Japan, aside from Rocket Slime and Swords.
In the end, its not just down to sales though. Dragon Quest is tied to a number of things that are not only popular, but have become tied to Japanese culture and the subconscious of a generation. Akira Toriyama, which many people identify simply as 'the guy who created Dragonball Z', actually became famous for creating both Dragon Quest and a Manga called Dr. Slump. He wouldn't even come to make Dragon Ball until after the first Dragon Quest was released. As well, much of the reason Japan loves RPGs and even gaming can be traced back to Dragon Quest. It was the game that got many people into gaming (of the people who have confirmed their influences to Dragon Quest are such as Hideo Kojima, Shigesato Itoi and Akihiro Hino). And even the music has become a major part of Japanese cultural identity, appearing on news broadcasts, sporting events and etc.
So really, there's a lot of compelling evidence to show that both series are popular. But when you bring together all the results of how much Japan likes Dragon Quest...well....its like comparing Mario to Mega Man sadly. Both are long standing and famous franchises...but one ultimately gets crowned 'popular'.
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