Can JavaScript itself be compiled to asm.js?
Not really, because of its dynamic nature. It's the same problem as when trying to compile it to C or even to native code - you actually would need to ship a VM with it to take care of those non-static aspects. At least, such a VM is possible:
js.js is a JavaScript interpreter in JavaScript. Instead of trying to create an interpreter from scratch, SpiderMonkey is compiled into LLVM and then emscripten translates the output into JavaScript.
But if asmjs code runs faster than regular JS, then it makes sense to compile JS to asmjs, no?
No. asm.js is a quite restricted subset of JS that can be easily translated to bytecode. Yet you first would need to break down all the advanced features of JS to that subset for getting this advantage - a quite complicated task imo. But JavaScript engines are designed and optimized to translate all those advanced features directly into bytecode - so why bother about an intermediate step like asm.js? Js.js claims to be around 200 times slower than "native" JS.
And what about non-statically-typed languages in general?
The slideshow talks about that from …Just C/C++? onwards. Specifically:
Entire C/C++ runtimes can be compiled and the original language interpreted with proper semantics, but this is not lightweight
Source-to-source compilers from such languages to JavaScript ignore semantic differences (for example, numeric types)
Actually, these languages depend on special VMs to be efficient
Source-to-source compilers for them lose out on the optimizations done in those VMs