Question: I was trying to transcribe my cat's noises into IPA. For example, her wanting to be petted is like [njæː] or [njaː]. But purring doesn't seem to fit any existing notation, except maybe [ʜ]. Has anyone ever mapped a cat's phonetic inventory? Their vocal tracts don't seem to be much different than ours.

Answer: 
Sure, why not? The main problem you will run into is establishing contrastiveness. I strongly recommend reading ch. 4 of the Handbook of the IPA. You can hear two different cats purring at the [wiki page](https://en.wikipedia.org/wiki/Purr), so the question is whether those are two different purr phonemes (technically known as "purreme") or one with a big variation in realization? You could transcribe it as [ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆], though you might have to explain that the breve implies a duration of about 5 msc. Apart from the inconvenience of writing all this junk, there is a deeper scientific question of whether it's wrong to imply that a purr is really made up of lots of tiny segments. So this is where, in dealing with language, somebody would propose a new symbol for this newly discovered sound. You could then propose [ﮛ] to represent this sound, but I doubt it would get approved.

With real language, the IPA symbol [a] represents a wide range of physical sounds, not a precise single physical sound (that doesn't even mean anything). At a certain point, an apparent "a" is far enough back that it sounds like the range of sounds represented as [ɑ], and then you can argue that writing "a" is an error and instead "ɑ" should be used. But that's only possible because there are languages that contrast [a] and [ɑ], so we know that [a] and [ɑ] are different things. The [a] of Arabic and the [a] of English (not US English) are different, but that doesn't motivate inventing new vowel letters: because, IPA is not a system for reducing arbitrary acoustic waveforms to combinations of letters.
I would like to encourage you in your transcription efforts on cat's phoneme inventory! I am a student of linguistics (B.Sc.) and I have two cats, too - that's why I have been thinking of doing the same one day. We often face men, even scientists, who claim that speech as a unique feature to the human race. When it comes up to explore speech in other species, the research is no longer free or open to new ideas but blocked as the results may question no less than the human supremacy: If we had evidence that various species do have language and use it on purpose (this requires advanced cognitive skills), it would be increasingly hard to keep up the human centric view on the world. So keep on noting down the distinctions that you recongnize in your cat's utterances and do not be discouraged if this would require creating a new kind of IPA chart for the felidae.