Moira Yip discusses this issue in Chapter 2 of her book Tone.
The short answer: four, possibly five.
HOWEVER, this question is not as straightforward to answer as one might think; tones can be defined in terms of phonological contrast--like phonemes--or phonetic contrasts--like phones. Just like segmental phonemes, tones are often analyzed as surfacing with contours that deviate from their true "underlying forms". For example, since the pitch contour of a syllable is directly affected by intonation as well as lexical tone, what might otherwise be a level tone might surface as falling in a declarative context and rising in an interrogative context. What's more, tones are often analyzed as undergoing phonological processes--known as tone sandhi--that are dependent on the identities of neighboring or nearby tones.
Let's take Mandarin as a familiar example. You stated that the third tone falls then rises. This is true of a carefully articulated syllable spoken in isolation (or sometimes at the end of a phrase), but actually in practice the third tone almost never really surfaces this way. If it is followed by another third-toned syllable, the pitch will basically just rise steadily. In other environments it often simply surfaces with a low, relatively level contour. In fact, some phonologists analyze Mandarin as having two level tones--high and low--and two contour tones--falling and rising. In such analyses the fall-rise surface contour of tone three is just an "allotone" of that "toneme" that occurs in a predictable environment, i.e. phrase-finally.
What about Cantonese? "Checked" syllables aside, Cantonese is usually analyzed by phonologists as having six contrastive tones. Three are usually considered level, two rising, and the last is either analyzed as "low-falling" or just very low level. I've made recordings of native speakers of Hong Kong Cantonese, and in that experimental context I found that that last tone usually surfaces (in a declarative context) with a contour that falls briefly but then plateaus and stays relatively level at the bottom of the speaker's pitch range. So is that a level tone? Who's to say??
Assuming one can decide which tones in an inventory are level, another challenge is deciding how many of them can be counted as contrastive. Imagine being out in the field with a new, unfamiliar language that appears to be tonal. You record a bunch of words, and there appear to be minimal pairs that suggest that whether a syllable is given a relatively high pitch or a relatively low pitch can affect the meaning of the word it's in. One straightforward analysis is to say that there are two contrastive tones in this language, "high" and "low". But another reasonable analysis is that some syllables are lexically marked for tone--we could call it "high"--and others aren't. All else being equal, syllables that are marked as "high" get realized with a higher pitch than those that aren't, which by default surface with a lower pitch. Likewise, a language that is observed as having three surface levels may be analyzed as having a "high" tone, a "mid" tone, and a "low" tone, but it may also be analyzed as having a "high" tone and a "low" tone, with unmarked syllables surfacing with a mid-range pitch. So, in theory, Yip notes, a system with n phonological tones can produce (n + 1) surface tones.
To summarize what Yip says about the typological facts:
"Underlying" three-tone systems are common: Huajuapan Mixtec, Nupe, Kunama, Punjabi. Most systems that are given "underlying" four-tone analyses--Grebo, Igede, Mambila, Mazateco, Chatino, Jianyang, Cantonese (?)--only display four surface tones. In theory, of course, the logical alternative is to assume "high", "mid", "low", and zero for these languages. Finally, there are languages that contrast five level tones on the surface, but "most are known only from fairly brief reports, with not enough detail to determine unequivocally whether there are actually five underlying level tones, or whether one or more are derived in some way." (Yip 27) These include Hei-Miao, Gaoba Dong, Dan, Trique, and Ngamambo Bamileke.
If you are interested in this topic, I highly recommend obtaining a copy of Yip's book!