A few casual runs I've done:
The Blackjack Run: get all 21 points on Grandpa's evaluation at the dawn of year 3. This was admittedly a lot easier before 1.6 adding mastery endgame recipes to the requirements of the "craft everything" point, let me tell you. 20 points would still be a flex.
Dry Run: Finish the community center without ever using the watering can. Because at the time I never set up sprinklers early enough, so this run I forced myself to do otherwise! For harder challenge, do it without Meadowlands for easy farming XP. (I originally did it in 1.5)
I Must Obey The Inscrutable Exhortations Of My Soul: Start with a random appearance. By the end of spring, decorate your house to match. On the last day of spring, pick up the decorative item or furniture from the traveling cart. By the end of summer, upgrade your house and decorate the new room to match the item purchased. On the last day of summer, etc. A fun little thing that makes you actually check out Pierre's wallpaper and flooring and Robin's furniture items.
Rip Van Winkle Run: speedrun completing the trash bear. Named for the two year nap you start the game with. Highly random: will the cooked gift requirements be locked behind an NPC at seven hearts, or available in the tavern? Play to find out!
The Corazon Run: Speedrun getting both the "Popular" achievement and the "Beloved Farmer" achievement. Nightmare mode: don't consult the wiki for NPC favorite gifts or schedules.