I have this:
That is 3 tenses, but a total of 4 5 4 = 13 "subcategories".
This boils down to more abstract concepts such as continuity and completeness. This is a nice table:
Could there be 16, 24, 32 different configurations, beyond my 13? If so, what are they at a glance? What is the general framework for thinking about all of them, taking cross-language thinking into account?
There are a lot of different things you can mark on a verb, including tense (setting a sort of temporal reference point), aspect (the relationship of the action to that reference point), mood (the relationship of the action to objective reality), and evidentiality (the source of your knowledge of the action). Some languages have these as separate, independent categories that you can mix and match; others have a certain fixed set of combinations you can choose from.
In Ancient Greek, there are three tenses (past, present, future) and three aspects (aoristic, imperfective, perfective). The aoristic aspect treats the action as a single point; the imperfective aspect indicates that it happened for a long time or was repeated or habitual; the perfective aspect means the aspect is over and done by the time of the reference point and the aftereffects are what matter. Present perfective, for example, corresponds to English "he has eaten" (so he's no longer hungry now): you're talking about the present, and what's relevant are the aftereffects of something that's over and done.
Almost all of these combinations of tense and aspect get their own special marking, though a couple are combined: present aoristic and present imperfective, for example, look the same. (Latin has the same set of three tenses and three aspects but combines them differently; in Latin the past aoristic and present perfective look the same, in Ancient Greek they don't.) But Ancient Greek also indicates mood—is this a real event, something possible, something hoped for, something commanded to happen, etc?—and in some of these moods, tense is not marked, only aspect. Commands, for example, only have aspect, not tense.
In Lingála, there's tense, aspect, and mood (no evidentiality), but only certain combinations are possible:
There's no such thing as a habitual subjunctive, for example; it's just not one of the combinations that exist.
In Hittite, there are only two aspects and no tense. You indicate whether the action has been completed or not, and that's it.
In English, you can combine aspects ("she has been working out" is talking about the aftereffects of something happening for a long time), but can't combine moods or tenses (*"she might can did that"). Syntactically, there's only one slot for the tense and mood, which is traditionally called the "T" or "I" position, but multiple slots for the aspect.
Actually, there can be a lot more tenses than the usual three, and also languages with no tense at all or only two tenses (past and non-past or, more rarely, future and non-future) exist. The Wikipedia article on Grammatical tense has some examples of additional tenses like Remote Past, Remote future, Hodiernal Past, Hodiernal Future (Past/Future, but today).
The subclassification inside the tenses are known as Grammatical Aspect, again, the languages of the world provide more possible aspects beyond English's simple, continuous, and perfect aspects. Examples of other types of aspects are completive and incompletive (denoting the completion or non-completionof an action), inceptive (beginning of an action), prospective (preparation of an action), and habitual Grammatical aspect is different from Lexical Aspect or Aktionsart that is determined by the semantics of the verb.
Finally, linguists often consider Tense, Aspect, Mood as an intertwined system of grammatical categories to be viewed together. These categories may be incremented by Evidentiallity and Polarity.
Postscript: You also may want to look at Reichenbach's theory of tense, introducing new tenses like the Posterior Past and Posterior Future.
