Well, you said personal, so personally, it's the Fantaisie-Impromptu for me.
* It came to me at the "right time" so to speak. I started learning it toward the latter part of high school, which was also around when I was starting to mature as a pianist. It was sort of the "big piece" that I played for my high school senior recital.
* When I was little, my mom used to play a cassette tape of various piano pieces as we drifted off to sleep. Fantaisie-Impromptu was one of them. For me, I associate the last few measures (where the left hand plays the cantabile melody) with a peaceful, drifting off to sleep sort of feeling.
* Now that I've come back to playing piano after over a decade of not really playing, I find that there are so many challenges with it, not just technical, but also musical, that I didn't notice back in high school; so many different ways to shape each phrase, and I'm not sure what sounds best. This keeps it interesting because I'll try to play it differently each time. It's the main piece that I can still play from my high school days, and the most polished.
Having said that, I think my main issue with the piece is twofold:
1) It's fairly repetitive; thus, when I play it, I play (mostly) the Fontana version for the first half, and (mostly) the Rubinstein version for the second half, to keep up the variety. But there isn't really much of an exposition type of thing (i.e. where the melody is restated in different ways, in a different key, etc.). On the flip side, I get bored of a lot of Beethoven's stuff because it often seems like he's just restating themes every which way possible and putting in filler instead of just selecting the best ones.
2) Having learned it and polished it, now I'm looking for something "more". I'm not sure what "more" means, but I guess sort of like a deluxe version of the piece or something. Hard to describe. (I feel the same way about Clair de Lune, where I like the sound, but wish there were sort of a higher-level version.) Because of this, I've been looking at Chopin's Bb Scherzo, although I'm kept busy with La Campanella, so it's unlikely I'll work on the next piece for a while.
But in terms of which Chopin piece has had the most impact on me personally, definitely the Fantaisie-Impromptu.
- I prefer the other impromptus (especially No.2 in F# minor) over the Fantasie-Impromptu, because of the ending. I think the end is boring. The piece reaches the climax already on the second page 
It's a simple A-B-A layout. It's overrated by many and not representative for the other works by Chopin. It's more and more becoming a popular popsong like Beethoven's Fur Elise.
Depends on how you play it and what you get out of the piece I guess. To me the climax is actually the top of the last page, i.e. after the octave run, before the left-hand melody part. The end is supposed to be tranquil, I don't know if you were hoping for fireworks or something. Maybe it depends on the person, but when I was looking for another piece to play, I listened to Chopin's other impromptus (including F# minor) and none of them struck out at me. (The Ab major was the closest one that I would've wanted to learn.) For me, the ending of the F# minor is what seems out of place, sounding like a wake-up call after the piece gradually faded.
- Overplayed by people who aren't ready yet. I know this argument is a bit lame, but I heard so many people play in a bad way. I can't stand kids with ages between 5 and 9 who play it and are obviously not ready for any romantic composer. If they want to be a great pianists they should stick to Mozart on that age. Campanella is another example. 95% isn't ready...
Overplayed is just indicative of how much people beyond a certain playing level are interested in that piece. I can easily compose a piece of any difficulty, the problem is whether or not it'll sound interesting enough that people would actually want to learn to play it; that it stands out among many other pieces of similar difficulty just speaks to how much it appeals to different people. Kids between 5 and 9 playing it is more indicative of their parents than anything else, not much different than kids with black belts. Realistically, anything that is popular is going to have lots that are done badly. Just because singing is popular doesn't mean everyone doing karaoke is going to sound good. (Many people want to go to Harvard. Just because a lot of people get rejected from Harvard, i.e. don't meet their standards, doesn't mean that Harvard is bad, nor should it reflect badly on Harvard.)
- It's plagiarism of Ignaz Moscheles's Impromptu in Eb major op.89 (1827). That's why Chopin didn't want to publish it in the first place. Wise man 
I think we should respect the decision of the artist and burn the piece 
I've never understood this theory, despite it being widely touted. Although the right hand melody bears a superficial similarity (in that the melody is composed of 16th notes that goes upwards and ends on an eighth note), the chord structure is different, the texture is completely different (the 4:3 polyrhythm for example), and where the respective sections fit into their respective pieces is also different. That several measures of this piece is similar in rough melodic shape to several measures of another piece seems like a poor excuse to call it plagiarism. There's plenty of other reasons that seem more plausible to me, such as its similarity to the Moonlight Sonata (mentioned above, I mean the melody notes are identical except an octave off to the Moonlight Sonata), and more importantly, that Chopin composed it on commission for someone and thus it wasn't his to publish. (Incidentally, what I hear in the frenetic pace of the Moscheles Impromptu is...Chopin's Minute Waltz.)
Topic: What is Chopin's greatest piece?