Speedster - Complete the game in 15 hours or less
It is suggested that you do this alongside Undefeated and Unchanging Armor on a Beginner Difficulty playthrough so you can use the Power Boosts, Defense Boosts, AP Boosts, and equipment (which come auto equipped) that you get at the start to make it easier for yourself.
There are two really important things to remember. The first is optional worlds do not need to be completed, these are;
100 Acre Woods
Olympus Coliseum (however completing this and the Hercules Tournament is recommended to improve Aero)
Atlantica OR Halloween Town (must complete one of the two)
The second is to use Aero/ra. This is defense and whilst it costs 2 MP is well worth it during later bosses.
On the runthrough you should be looking at completing most worlds in around 40 minutes, with Traverse Town 1 and 2 taking about 20 minutes, Monstro and Neverland taking 30 minutes. Overall, Hollow Bastion and End of the World can take an hour or two each.
It's best to choose to start your journey at the dead of night by selecting mainly the third options when asked by Tidus, Selphie, and Wakka about your hopes and fears.
A good timeline to follow, allowing for some time delays (and late game leveling if needed) is;
Destiny Isles finished at 00:40
Traverse Town 1 finished at 01:10 (take the top gummi route)
Wonderland finished at 02:00
Deep Forest finished at 03:00 (take the bottom gummi route back toward Traverse Town)
Traverse Town 2 / World Set 1 finished at 03:30
Agrabah finished at 04:20 (take the top gummi route)
Monstro finished at 05:00
Halloween Town finished at 05:50
Neverland finished at 06:30
(at this point the Hercules Cup should be available in Olympus Colosseum)
Traverse Town 3 / World Set 2 finished at 06:40
Olympus Colosseum finished at 7:10 (do the story and the Hercules Cup, use trinity on the block, go to Neverland and use Yellow trinity in the hold to open the door and get an Aero upgrade, better to do this before finishing HB's first run so you're facing lower level Heartless)
Hollow Bastion 1 finished at 08:40
Traverse Town 4 finished at 09:00 (collect postcards and hand them in, gives a Defense Boost, as well as use Blizzard on the candles in 1st District for another Defense Boost)
Hollow Bastion 2 finished at 9:20 (Talk to Aerith in HB 3 times for Cure upgrade)
World Terminus Save Point at 10:00
Any level grinding you wish to do and reach the Final Save Point by 13:00
Final Battle Start by 13:30
The final battle chain can take OVER an hour and as such it's best to have started the final fight by 13:30. If any late game (Hollow Bastion or End of the World) battles are challenging, don't forget you can summon Tinkerbell. Unlike other summons she'll stick around as basically a constant Regen and a once per battle Auto-Life [and then leave]. This can even be done during End of the World battles by landing on the Bosses and summoning. Remember that you need two teammates to summon.
Or you could just wait till the timer resets after 99 hours 59 minutes and 59 seconds if you are also going for the Level Master trophy