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The IPA's current tone system can show five different tone levels, and any contours formed from them.
Is there any language for which this is insufficient? In other words, is there any (known, natural, attested) language which uses six or more phonemically-distinct tone levels?
I don't think there are any attested languages that require more than five (5) phonemic levels of pitch to describe. However, there is one language Cori with six (6) surface pitch realizations, although it can be analyzed with a tone inventory of three level tones. I'm looking for some more non-Wikipedia sources and a better explanation of the situation in Cori, but here's what I have so far.
Bench has five level tones and one rising tone 23. Here's a source that Wikipedia cites.
Cori can be described as having six phonemic levels of pitch, apparently. However, 2 is an allotone of 4 and 3 and 5 are surface realization of sequences of tones. So, a minimal analysis only needs a three tone inventory.
The situation in Chori exemplifies a widespread problem with discussions of allophony, that "allophone" has different meanings. The classic definition of allophone is that two phonemes (or, tonemes) are allophones iff the variants appear in surface-complementary environments. This is not the case in Chori: all 6 tones are surface-contrastive. The alternative definition is "you could remove some of the tones from underlying forms by applying some set of rules". The data-source, Dihoff, pursues a toneme-minimizing analysis where 1→3/_6, 4→2/_1 and 2→5/_6. The particular rules are well-enough motivated since they apply in phrasal contexts, but the underlying representations are not always motivated. Dihoff notes that these rules may be limited to certain grammatical constructions, for example in the present tense, /1/ becomes [6] after a 3s subject and /4,6/ become [3] after a 1s subject. This analysis may be technically tenable if you decompose any [3] into an abstract sequence /1+6/, but this is entirely contrary to the spirit of the traditional concept "allophone" which is based on distribution in surface forms, not underlying forms. Dihoff's analysis only assumes that you can remove tones /2,3,5/ from lexical entries, it does not assume that you can remove those tones from derived representations, including the surface form. In fact, his discussion starts with a tone-minimizing analysis with just /1,4,6/ but morphs into a full 6-level analysis once he gets to talking about tense-inflection and the like.
IPA allows concatenation of level markers, using either accents or bar-markers, so that contours are irrelevant – you can have rising, falling, high-rising and so on as contours. The only thing that would pose a problem for IPA is a 6th level. As it happens, IPA sort of failed in its remit to specify standard letter-shapes for contours. Given 5 levels, there are very many contour tones than can be created (and that are attested) such as extra-high-to-extra-low vs. extra-high-to-mid vs high-to-extra low etc. But they only provide 5 letters for tone contours, and they leave it to the author to come up with a clever way to indicate more subtle contours. This means that authors dealing with language that have many levels and many contours formed from those levels have to operate outside the IPA system, and use one of the two (mutually incompatible) non-IPA numeric-superscript notational systems.
I don't see Vietnamese nặng (low glottalised) or ngã (rising broken) among the IPA symbols. Vietnamese tone pitches and contours cannot be separated from phonation type or manner of articulation of final consonants, so classification based on pitch is not adequate. See Vietnamese Tone: A New Analysis (Pham, 2004).
