What do you think first drew Applejack to Twilight, and what caused them to become so close?

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Despite being from Canterlot and being royalty, Twilight is one of the most down-to-earth and genuine ponies in Equestria.

I think at their very first meeting, Applejack was mostly trying to be polite to the Canterlotian Inspection Pony and impress her with all the food.

Later on when Nightmare Moon arrived, Applejack was the first one in the group to recognise that Twilight had some idea of what was going on and encourage her to share what she knew. Applejack is generally a friendly pony, and she probably decided that this poor unicorn just needed a little bit of reassurance to get going with the whole trying to stop the Nightmare thing.

Interestingly - Applejack's Element task was the only one of the tasks that was solely focused on Twilight.
Everypony else's tasks affected either external beings (Fluttershy earning Kindness with the Manticore, Rarity earning Generosity with Steven) or the group as a whole (Pinkie earning Laughter with her song to the group, Rainbow earning Loyalty by not being willing to leave the group behind, Twilight earning Magic for her realisation).

But for AJ's task to earn Honesty? None of the other ponies were in danger, Applejack wasn't asking anypony else to believe in her - it was all about that first connection of trust between her and Twilight. It's something that's been maintained between them ever since. The early seasons especially seemed to highlight it a lot - the first time we see Twilight hanging out with one of her new friends in The Ticket Master, it's Applejack. The next episode is Applebuck Season. Applejack saved Twilight again in Dragonshy. They had the argument & make-up in Winter Wrap-up etc etc.

If Twilight's not there, it's most often AJ who takes over to direct the group, and she's the one who speaks to Twilight the most, on behalf of the others. Especially in all the season openers/endings. Seriously.

Tl;dr: the strenuous circumstances around Nightmare Moon's return forged a strong bond between them based explicitly on care and trust, making it one of the stronger friendships within the group for at least the first few seasons.