Is it possible to compile TypeScript into minified code?

The TypeScript compiler does not support the generation of minifies or obfuscated code. you will need to use another tool against the JavaScript output of the compiler.

There's an open feature request: microsoft/TypeScript#8

I am the author of typescript-closure-compiler.
Basically it is a tool that emits files that are compatible with google closure compiler, which is probably the best minifier/optimizer/obfuscator out there.
I've also created a playground that can show you the generated output.
You can take a look at an example project which uses this tool.
The minification process is done using gulp.