How many transistors does an 8 x 1 multiplexer have?

Look at the logic diagrams here and consider that each input to each gate requires (at least) two transistors.

http://www.nxp.com/documents/data_sheet/74HC_HCT15...

I make it about 15x2 + 8x5x2 + 8x2 + 4 = ~130

This assume you are talking about a digital mux and CMOS logic,

16 for the multiplexer itself, two for each channel, for a simple one.

Real ones like DG588 have 6 transistors per channel plus about 50 for the logic, so about 100 transistors