Does an uncompressable string exist? [closed]

Question

I was wondering if there is one or more strings that cannot be losslessy compressed. More formally:

Let String be a string, f(var) a compression function which returns a compressed version of var, g(var) a decompression function such that g(f(var)) = var and strlen(var) a function which returns the length of var,
is there a valid value for String such that strlen(String) < strlen(f(String)) or strlen(String) = strlen(f(String))?

Theoretical answers are welcome, as well as examples in different languages and with different compression algorithms.

Answer 1

The pigeonhole principle tells us that for any given compression function^*, there must always be at least one input string that will be expanded.

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_{* i.e. a function that genuinely compresses at least one input string.}

Answer 2

I would expect that this string would fit the bill: ""