FG&RP boasts the prestigious title of having the most posts in any board; 941244. It's closely followed up by GD, with 895963 posts. The next highest posts/board is Other Games, at a (comparatively) low 350568 posts.
When it comes to 'busiest', the best metric I can think of to measure that is posts/day in that specific board. Since, sadly,
my favorite part of the forum doesn't record posts/day by board, we'll have to take the average. The best solution I could think of is to find the first post in each board to determine its lifespan, and then divide the number of posts by that number.
That means GD is 278 posts/day, while FG&RP tops it out at 308.
FG&RP also has more topics as opposed to GD, so I think we can safely say that FG&RP is the busiest, but it should be noted that the Happy Thread is the single largest and most viewed thread on the forum.