Process pineapple in a blender or food processor until pureed.
Scrape the puree into a pan and add the superfine sugar; cook about 5 minutes until reduced by 1/3 (mixture should be thick, but not as thick as preserves so you can add a little water to thin out).
Meanwhile, make the caramel by placing the granulated sugar in a heavy pan and cook over medium heat until the edges start to caramelize; shake pan, do not stir.
Remove from heat once all the sugar has dissolved and the sugar is golden brown.
Carefully stir in the lime juice (it will sputter and spit).
Divide the caramel among 6 ramekins and turn them so caramel distributes evenly.
Preheat the oven to 350 degrees F.
Stir the eggs into the cooled pineapple mixture; divide egg mixture evenly among the ramekins.
Place ramekins in a roasting pan and pour warm water into the roasting pan so that it comes halfway up the sides of the ramekins; cover with aluminum foil and bake 45 minutes until set; let cool.
Invert ramekins onto dessert plates (loosen sides with a knife, then place dessert plate over the ramekin and flip over).