Huge amounts. Practical scala apps leverage huge parts of the Java infrastructure. The Scala types are often thin wrappers over the Java types, JDBC, NIO, etc. It'd be a bit like trying to write a C++ app without access to, say, the STL, or Boost.
Scala and the JVM are very tightly coupled - targeting LLVM or something should be possible, but getting that to run with high performance would be very difficult.