So I preface this with a claim: any language that has a fixed semantics that can express computable functions is implementable. I'm hedging my bets here by not saying specifically what I mean by the "semantics", but I generally mean something that is either an operational semantics (big or small step), a denotational semantics, or axiomatic semantics.
So, with that out of the way, there remain two questions: Are there languages which don't have any semantics at all (and are therefore not implementable by any of the usual means), or are there languages which haven't been implemented, despite it being possible to do so?
The latter case is obviously true, but it's not very interesting. Just because people dream up a language doesn't mean it's a good idea. I could think of a language right now, and it wouldn't have a compiler or an interpreter. Not very interesting.
The former case is a little more interesting. There are languages that are actually logics and (via the Curry-Howard correspondence) proofs in these logics are also programs (and vice-versa). If the logic is not decidable, or if the language includes some kind of higher-order constructs that are not strongly normalising (such as some very expressive type system), then compilation or interpretation becomes a slightly scary affair. This might start to verge into the territory of "not compiling or interpreting", but that's a slightly spurious claim, because if we can make a computer manipulate these symbols, we can write an interpreter. Similarly, if it has no semantics, then it's probably not operating over computable functions - it's probably just natural language of some kind at that point, no longer a "programming language".
This is especially true because we can argue that the Church-Turing thesis says that it will always be possible to write an interpreter for a Turing-complete language, even if that interpreter ends up being another Turing machine.
I see some other answers have made claims about machine language. There is nothing magical about machine language except that it has a very compact representation. One could very easily write an interpreter for it, and many exist (e.g. the various virtual machines that faithfully implement the execution semantics of a hardware processor in software).
The languages that meet your criteria are likely to be ones that have no execution semantics, i.e. they do not express computable functions. Here are a couple languages with fixed semantics I know of that are not directly executable by either an interpreter or by compiling them to anything else without enriching these programs with additional information: