You ask a loaded question. C# is merely a language and does not require the .NET Framework. The process of compiling it requires a compiler, which may or may not itself take a dependency on the .NET Framework (Microsoft's C# compiler does not -- it is written in native code). Your program will need to reference some assembly where types, classes, and methods can be found for your use. You can remove system.dll and mscorlib.dll from your references list and reference your own assemblies. So you can avoid dependencies on the .NET Framework if you really work at it. But in the end, unless you have a C# compiler that compiles programs to native code you still have a dependency on the CLR.
That's a very technical way of saying... almost nothing. But it answers your question. :) More practically useful however is how to get your C# programs to run with a minimum of dependencies. mkbundle from mono will actually let you compile it all into an .exe with virtually no dependencies.
But if you want to stick with the Microsoft .NET Framework, you can achieve a much lighter footprint and faster install of the dependencies you commonly need by using the Client profile of .NET 3.5 SP1. You can read about it here:
http://msdn.microsoft.com/en-us/library/cc656912.aspx
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