Do all words have a part of speech? The closest counterexample I can think of is yes. The dictionary says its supposed to be an adverb but it doesn't really strike me as something that modifies a verb. Are there any words in common use that simply don't have a part of speech? Or, alternatively, are there any parts of speech that they just don't teach in school because they're too abstract to the children learning their parts of speech?
Learn English – Do all words have a part of speech
grammarparts-of-speech
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One problem is that the entire concept of "part of speech" is very old. How we use it in English, especially in dictionaries, goes back to the study of Latin and Greek. In this view of English grammar "adverb" is the catch-all category where everything that doesn't fit into one of the other traditional categories ends up. (The others being noun, verb, adjective, pronoun, preposition, conjunction, and interjection.)
Now there is no one, true description of any language (except perhaps constructed languages such as yours). There are merely alternative or competing descriptions which appear over time as more independent analyses of the language are undertaken. Such descriptions or analyses may be called "grammars".
Most (but not all) grammars include a concept of word class under one name or another. So one problem is that "part of speech" has two meanings. One is the specific set of eight categories from the classical languages, the other is as a synonym for word class, which is a lot looser.
So all your example words are adverbs under this older stricter view of parts-of-speech, but their qualities and quirks can be much more thoroughly investigated in newer ways. And various new ways will have various new terms for the classes they put these various words into.
Unless you are inventing a new language specifically to embrace the classical parts of speech you don't have to worry in which they belong, but if you are inventing a new language to learn more about how language works then it will be worth your time reading up on the many newer grammars and language descriptions and analyses.
In this instance, "least" is a noun meaning "smallest, lowest, or minimal amount."
Rephrasing the original sentence to demonstrate this meaning gives us: "Its lowest speed is faster than sound." Rephrasing Fumble Fingers's example sentence gives us: "A ton is its lowest possible weight," or, "The smallest weight it could have is a ton."
The rephrasing also helps clarify a more subtle point about the original sentence in question here. "At least" is taken as definitive, precise, and indicative of a specific point in a scale of measurement. So if the scale of measurement is speed, we might say "at least 700 miles per hour," or "at least the speed of sound." We would not usually say, "at least faster than the speed of sound," just as we would not usually say, "at least more than 700 miles per hour." "Faster" and "more than" are not defined points in the scale of measurement, and so they eradicate the functional bottom limit implied by "least." If something moves "at least faster than sound," its minimal speed can be any speed whatsoever as long as it is faster than the speed of sound.
One could argue that it is possible and reasonable to say "at least faster than sound," or in other words, "its slowest speed is nevertheless faster than the speed of sound." And I can't argue that this phrasing is impossible or non-reasonable. What I can argue, however, is that this phrasing is slightly awkward, and inconsistent with the well-accepted and useful intended meaning of "least." I would then say it is preferred for precision, effectiveness, and clarity of expression to use a single, specific point in a scale of measurement when you are referring to "the least."
Best Answer
It is intended that all words have a part of speech.
As for your second question - "are there parts of speech they don't just teach in school?" Traditionally, The eight parts of speech are: nouns, verbs, pronouns, adjectives, adverbs, prepositions, conjunctions, and interjections. Students learn all of these.
However, depending on what we're doing, we can use a lot more different types of speech. For example, for my natural language processing project, I used 44 different types of speech (including some like "numbers" and "periods")