Silicon lifeforms have been speculated.
HoweverBut......
Silicone-Based Life
Silicon life would probably be based on Silicones, polymers of alternating silicon and oxygen. While in general silicones are less stable than hydrocarbons, they would have an advantage in certain environments. In particular, in an atmosphere or environment where sulfuric acid was common, silicones would have greater resilience than carbon-based molecules.
Silicones do run into some troubles, however. Namely, silicon that gets exposed to oxygen often forms silicon dioxide. Since it would be the analog of carbon dioxide, there's reason to believe such life would exhale it as part of respiration - breathing out dust and sand. Silicon dioxide is a solid at most temperatures, and not water-soluble, so you'd have problems removing this waste from cells, recycling biological material, etc. Now, "life may find a way", and in the case of silicone-life it's possible the solution would be ammonia. See Ammonia as Biological Solvent for a discussion of how it might work.
So, the environments most friendly to Silicon-based life are rich in ammonia and sulfuric acid. Ammonia's boiling point is below water's freezing point, so the planet would either be very cold or have extreme pressure (because at higher pressures, the boiling point of ammonia is much higher) - say, 60 atmospheres. Ice Giants and places like Titan, moon of Saturn might be viable habitats for silicon-based life. In other words, any planet likely to be hospitable to silicon life isn't going to be nice for humans. Maybe, just maybe, you might be able to get around that by having a Cloud Planet scenario - a layered Gas Giant that has some breathable earth-atmosphere at the high altitudes and a high-pressure silicone-ammonia environment further down. The notion of shared environments is iffy, to say the least.
Molten Silicon Life
Alternately, since silicon structures survive great temperatures, it's possible silicon life might exist in places of extreme heat. These thermophile / extremophile lifeforms would be made of molten silicon. This version of silicon life might exist deep within planetary cores. They may even form a shadow biosphere inside our own planet, of which we are currently ignorant. They may draw energy from infraradiated heat (see Non-Green Photosynthesizers) or from the oxidization of iron. Nothing I've read really suggests what would be used as a solvent by such creatures - but I assume it'd probably be some extremely hot metal that's normally a solid on the surface of the earth.
Obviously, the prospects of humans interacting with molten silicon life are no more promising than our capacity to interact with super-chilled silicone-ammonia life.
The young have aspirations that never come to pass, the old have reminiscences of what never happened.