Not really: you have to keep in mind the amount of expertise and ressources that already went into silicon, as well as the geopolitics and sheer availability of silicon. The closest currently available competitor is probably gallium arsenide. That has a couple of disadvantages compared to silicon

• It's more expensive (both due to economies of scale and the fact that silicon is just much more abundant in general)
• GaAs crystals are less stable, leading to smaller boules.
• GaAs is a worse thermal conductor
• GaAs has no native "oxide" (compare to SiO₂) which can be directly used as an insulator
• GaAs mobilities are worse (Si is 500 vs GaAs 400), which means P channel FETs are naturally slower in GaAs, which makes CMOS structures impossible
• GaAs is not a pure element, which means you get into trouble with mixing the elements
  You usually see GaAs combined with germanium substrates for solar panels, but rarely independently of that (GaAs is simply bad for logic circuits).
  In short: It's not really useful for logic gates.

Germanium itself is another potential candidate, especially since it can be alloyed with silicon which makes it interesting from an integration point-of-view.
SiGe is very interesting from a logic POV considering its high forward and low reverse gain, which makes it interesting for low-current high-frequency applications. Since you naturally have heterojunctions which allow you to tune the band-gap (on the other hand you get the same problem as in GaAs: it's not a pure element so you need to tune the band-gap).
One problem specifically for mosfets is the fact that you don't get stable silicon-germanium oxides, which means you can't use the established silicon-on-insulator techniques.
Cost is also a limiting factor: before even starting to grow crystals you have the pure material cost, which is roughly $10/kg for silicon, and $800/ kg for germanium.
That's why, despite the fact that the early semiconductors all relied on germanium, germanium based systems never really became practical: It's harder to do mass production, and even if you can start mass production it will be very expensive (that's why if you do see germanium based tech, it's usually in low-production runs for high cost specialised components)

There's some research going on in commercialising these techniques but that's still years away.