This wikepedia page defines c++ as a "white space independent language". While mostly true as with all languages there are exceptions to the rule. The only one I can think of at the moment is this:
vector<vector<double> >
Must have a space otherwise the compiler interprets the >> as a stream operator. What other ones are around. It would be interesting to compile a list of the exceptions.
Following that logic, you can use any two-character lexeme to produce such "exceptions" to the rule. For example, += and + = would be interpreted differently. I wouldn't call them exceptions though. In C++ in many contexts "no space at all" is quite different from "one or more spaces". When someone says that C++ is space-independent they usually mean that "one space" in C++ is typically the same as "more than one space".
This is reflected in the language specification, which states (see 2.1/1) that at phase 3 of translation the implementation is allowed to replace sequences of multiple whitespace characters with one space character.
The syntax and semantic rules for parsing C++ are indeed quite complex (I'm trying to be nice, I think one is authorized to say "a mess"). Proof of this fact is that for YEARS different compiler authors where just arguing on what was legal C++ and what it was not.
In C++ for example you may need to parse an unbounded number of tokens before deciding what is the semantic meaning of the first of them (the dreaded "most vexing parse rule" that also often bites newcomers).
Your objection IMO however doesn't really make sense... for example ++ has a different meaning from + +, and in Pascal begin is not the same as beg in. Does this make Pascal a space-dependent language? Is there any space-independent language (except brainf*ck)?
The only problem about C++03 >>/> > is that this mistake when typing was very common so they decided to add even more complexity to the language definition to solve this issue in C++11.
The cases in which one whitespace instead of more whitespaces can make a difference (something that differentiates space-dependent languages and that however plays no role in the > > / >> case) are indeed few:
inside double-quoted strings (but everyone wants that and every language that supports string literals that I know does the same)
inside single quotes (the same, even if something that not many C++ programmers know is that there can be more that one char inside single quotes)
in the preprocessor directives because they work on a line basis (newline is a whitespace and it makes a difference there)
in line continuation as noticed by stefanv: to continue a single line you can put a backslash right before a newline and in that case the language will ignore both characters (you can do this even in the middle of an identifier, even if the typical use is just to make long preprocessor macros readable). If you put other whitespace characters after the backslash and before the newline however the line continuation is not recognized (some compiler accepts it anyway and simply checks if last non-whitespace of a line is a backslash). Line continuation can also be specified using trigraph equivalent ??/ of backslash (any reasonable compiler should IMO emit a warning when finding a trigraph as they most probably were not indented by the programmer).
inside single-line comments // because also there adding a newline to other whitespaces in the middle of a comment makes a difference
Like it or not, but macro's are also part of C++ and multi-line macro's should be separated with a backslash followed by EOL, no whitespace should be in between the backslash and the EOL.
Not a big issue, but still a whitespace exception.
This is because of limitations in the parser pre c++11 this is no longer the case.
The reason being that it was hard to parse >> as end of a template compared to operator >>
While C++03 did interpret >> as the shift operator in all cases (which was overridden for use in streams, but it's still the shift operator), the language parser in C++11 will now attempt to close a brace when reasonable.
set<set<int> >.' '." ".else return x;, void foo(){}, etc.
