In La Campanella by Liszt, in the running passage which I added a photo off, shouldn't the highest note (which I marked with an arrow) be a G because the F double sharp gets carried over? In all of the recordings I listened to, it's played as an F sharp. What is the correct way to play it?
While accidentals will often prompt publishers to put courtesy natural signs before the note's pendants in another octave if they occur later in the same bar, this is not actually necessary: unlike the signs of a key signature, accidentals only apply to the exact line they're written in. The 𝄪 does not apply to another F♯ but the one it's actually written before.
Apart from that... even if the double-sharp carried over, it would not be a G, even though F𝄪 is indeed played with the same piano key.
Nevertheless, watch out... as Richard said, this is not something that's always handled consistently. But it is the official convention.
Composers (and editors, and publishers, etc.) vary on whether or not an accidental on a particular pitch (ie, this exact F♯) applies to all members of that pitch class (ie, all F♯s). However, the fact that they specified a high E♮ tells me that this could be an F-doublesharp.
But I really doubt it should be an F-doublesharp. In fact, if I were playing it, I would almost think it was an F♮ instead of an F♯, since it's just a big chromatic scale. But the score shows an F♯; without any proof from a Liszt manuscript or anything like that, it's an F♯.