What is a "hemidemisemiquaver"?

It is a 64th note (the 64th note of a song you play).

https://en.wikipedia.org/wiki/Sixty-fourth_note

The basic unit of time in music is represented by a semibreve. "Semi" means "half", so a semibreve lasts for half as long as a breve. However, the breve is now obsolete and the semibreve has taken over. Typically, a semibreve lasts for 4 beats.

Other musical notes are derived from the semibreve by halving. Thus a minim lasts half as long as a semibreve (or, to put it another way, two minims last as long as a semibreve).

Likewise, a crotchet lasts half as long as a minim so 4 crotchets last as long as a semibreve.

Dividing by two again give us the quaver, lasting for one-eighth of a semibreve.

Now "hemi", "demi" and "semi" are prefixes meaning "half" (think hemisphere, demigod and semi-detached). Thus a semiquaver would last one-sixteenth of a semibreve, a demisemiquaver for one-thirtysecond and therefore (here's the answer at last) a hemidemisemiquaver for one-sixtyfourth of a semibreve.

https://en.wikipedia.org/wiki/Whole_note
https://en.wikipedia.org/wiki/Quaver
https://en.wikipedia.org/wiki/Sixty-fourth_note
https://en.wikipedia.org/wiki/Musical_notation

But that does not mean it's necessarily the 64th note of a song, it could occur anywhere in a piece. As I understand American music terminology, calling a note a 64th note means, as Watch says, 1/64th of a full note. Probably easier to learn than all the hemis and demis we use on this side of the pond.