After watching videos like the ones from Language Stuff, I'm wondering how fast anyone has learnt Esperanto to a level where they're able to hold a conversation (and this is documented by a video, ideally).
Chuck Smith has interviewed a girl, who had finished the Duolingo Esperanto tree in 3 days. You can see the video here. Although she is not exactly fluent in this video. There is also another video interview with a different girl who had finished the tree in 6 days. See it here. She is definitely better than the first one which can be expected — she has spent twice the time in learning! :)
While they both can hold a conversation, it can be considered a beginner one, and it all depends on your criteria for "conversation". If you are asking about the fastest one can reach the level of discussing fluently about the metaphysical concepts of the universe, then I believe that would probably take more than 3 or 6 days.
Leo Tolstoy claimed to have gained the ability to "freely" read Esperanto texts after just two hours study.
Earlier this month (Sept 7) I had a video chat with a young lady who claimed to have been learning Esperanto for nine days. I did not film it, but for sure she was able to "hold a conversation" with me. (Like Tolstoy, she was clearly an experienced language learner.)
I also remember being impressed with this series, in which Benny presents highlights of his girlfriend's Esperanto-learning journey. She studied for an hour a day and he filmed every moment of her study.