Or, to be a little more precise: which programming languages are defined by a context-free grammar?
From what I gather C++ is not context-free due to things like macros and templates. My gut tells me that functional languages might be context free, but I don't have any hard data to back that up with.
Extra rep for concise examples :-)
Almost all modern languages are context free. C++ is one of the rare exceptions due to some things you can do with templates. No others come to mind that are not context free, although things like XML integration in VB.NET and also LINQ queries in both C# and VB might require context sensitivity. At least, Paul Vick, then head of the VB team, mentioned that XML in VB is quite hard to parse.
The set of programs that are syntactically correct is context-free for almost all languages.
The set of programs that compile is not context-free for almost all languages. For example, if the set of all compiling C programs were context free, then by intersecting with a regular language (also known as a regex), the set of all compiling C programs that match
^int main\(void\) { int a+; a+ = a+; return 0; }$
would be context-free, but this is clearly isomorphic to the language a^kba^kba^k, which is well-known not to be context-free.
VHDL is somewhat context sensitive.
(Google: parsing-vhdl-is-very-hard)
To go for the most dramatic example of a non-context-free grammar, Perl's grammar is, as I understand it, turing-complete.