This dissection by Noam D. Elkies uses 8 pieces:

According to Elkies, it was JHI

(identity unknown,
perhaps connected with the Jeremiah Horrocks Institute) who first
showed that 8 pieces are necessary.
The proof is simple: the 6×6×6 cube has 8 corner cells.
Any piece big enough to include two of the corner cells will be too big
to fit in a smaller cube.
Therefore each of the 8 corners of the 6×6×6 cube
must belong to a separate piece.

The earliest 8-piece dissection known to Elkies
appears in Harry Lindgren's
*Geometric Dissections* (1981).
The book attributes it to R. F. Wheeler.
It is more complicated than Elkies's.

It is not known how many of the pieces in an 8-piece dissection
can be rectangular prisms.
However, T. H. O'Beirne found a dissection with 9 pieces,
all rectangular prisms.
See Martin Gardner's *Knotted Doughnuts
and Other Mathematical Entertainments* (1986).

Edo Timmermans analyzed the problem and found a solution in which no pieces are rotated. See this page of Greg Frederickson's.

*Last revised 2021-06-05.*

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