Similar to milyen/olyan/ilyen, these three words are used to describe things – but in this case, it’s more about expressing quantity (i.e. how much/many of something).
mennyi: how much
One of the most common ways you might see mennyi used is in the phrase “mennyibe kerül?” (how much does it cost?)
Ez a ruha mennyibe kerül? (How much does this dress cost?)
Mennyi egy euro? (How much is a Euro?) – i.e. how much is a Euro in comparison to another currency
Mennyi volt a számla? (How much was the bill?)
Mennyi idő a bankból a postára menni? (How much time/how long does it take to go from the bank to the post office?)
Mennyi vs Hány: How Much vs How Many
Both words are used to express the quantity of something, but mennyi is less particular about how the quantity is measured, and is used when the amount can be expressed in less specific amounts.
Hány, on the other hand, is asking for a more exact amount/quantity, and is used when an actual count or measurement is required. For the most part, it’s the same difference as in English between how much and how many. Some things only sound right with “how much“, and some only sound right with “how many“. For example, you wouldn’t say “how much cats do you have?”, but “how many cats do you have?”
Mennyi sört kérsz? (How much beer do you want?)
Hány pohár sört kérsz? (How many glasses of beer do you want?)
In the first sentence, we are asking in general how much beer the person wants. They could say “a glass”, “a pitcher”, or “enough to fill up an SUV”.
In the second sentence, we are asking specifically how many glasses of beer they want. So since we are actually counting something (glasses), hány is more appropriate here.
Mennyi ideig1 tart a gyűlés? (How long (lit: how much time) will the meeting last?)
Hány óraig tart a gyűlés? (How many hours will the meeting last?)
1-ig is a suffix that means “until”, and is used with certain time expressions. You can read about it in the cases overview (and I will probably make a more in-depth post on it at some point)
Again, in the first sentence we are just asking about time in general, so mennyi is appropriate. A person could answer with minutes, seconds, hours, etc. But in the second sentence, we are specifically asking for the amount of hours, therefore we use hány.
Annyi / Ennyi
annyi = that much, so many/much (of that)
ennyi = this much, so many/much (of this)
These work similar to olyan/ilyen, except annyi/ennyi are used to emphasis quantity (how much) rather than quality (what kind/what it’s like). While sometimes the difference is more nuanced, the general rule is that olyan/annyi is about things out of sight, distant, in the past – i.e. something not present or visible at the moment. Ilyen/ennyi is more about what you can see right at the moment – something visible or perceptible, or previously mentioned.
Note that annyi is usually followed by a second clause with more information. (i.e. “there are/were so much/many (x), that…”)
Annyi ember volt a boltban, hogy alig tudtam keresztülverekedni magam (There were so many people in the store, that I could barely fight my way through)
Annyi pók van a padláson, hogy nem merek felmenni (There are so many spiders in the attic, that I don’t dare go up there)
Nem tudtam, hogy ennyi munka lesz (I didn’t know (it) was going to be this much work) – often said to myself, when I have the sudden urge to deep clean the house and then promptly run out of steam 1/3 of the way through.
Mennyire/Annyira/Ennyire vs Milyen/Olyan/Ilyen
To express to what extent something is (i.e. emotions, feelings, desires), you can add -ra/-re to the end of mennyi, annyi, or ennyi. So they become mennyire, annyira, and ennyire.
Mennyire akarsz Magyarországra utazni? (How much do you want to go to Hungary?)
Annyira akartam Finnországba menni, de most nem tudok, mert beteg vagyok (I wanted to go to Finland so much, but now I can’t because I’m sick)
A jövő hónapban megyek tanulni Europába. Soha nem voltam ennyire boldog! (Next month I’m going to study in Europe. I’ve never been so/this happy!)
Differences
Milyen/Olyan/Ilyen can be used similarly to express extent, but as mentioned previously, they are more qualitative (descriptive). Mennyire/Annyira/Ennyire, on the other hand, are more quantitative (emphasizing the AMOUNT of something). In some cases you can use them interchangeably, and the “correct” one sometimes just depends on the nuance of the sentence. You can use either word and you’ll be understood, even if it’s not the most “fitting” one. It’s another thing that you will pick up more easily as you advance in your studies and encounter more examples.
Mennyire fontos ez? (How important is this?)
literally: of how much importance is this?
Mennyire vagy boldog attól, hogy férjhez megy a lányod? (How happy are you now that your daughter is getting married?)
literally: how much happy are you… (it sounds weird in English, but it’s specifying the amount of happiness, so mennyire is a slightly more fitting choice).
Compare
Annyira boldog vagyok, hogy látlak (I am so happy to see you)
literally: I am this much happy to see you
vs
Olyan1 boldog vagyok, hogy látlak (I am so happy to see you)
literally: I am this kind of happy to see you / I am happy like this to see you
1Olyan typically corresponds to “like that“, and refers to things that are more distant/not visible or present. Ilyen typically refers to things immediately visible and apparent. While it might seem like this sentence should take ilyen, unless you are doing something specifically to show what your current “happy” looks like (i.e. pointing to a happy meter gauge nearby or something), then olyan is more appropriate. The same applies to annyira/ennyire.
Both sentences are equally correct, and the difference is very subtle. Both mean that you are very happy to see the person, and you can choose to express that how you wish.
Compare
Jó látni, hogy ennyire boldog vagy (It’s good to see you so happy)
vs
Jó látni, hogy ilyen boldog vagy (It’s good to see you so happy)
Again, the difference here is very subtle. It’s kind of like “it’s good to see you this amount of happy” vs “it’s good to see you happy in this way“. Both are correct and acceptable.
Unlike the last comparison, you can use ilyen/ennyire here, because you are referring to a person you can see who is clearly showing their mood, and you can tell that they are happy.
In Summary
Don’t worry too much if you mix these up, as you will be understood regardless. Practice and exposure will help you get more familiar with the subtler nuances between them.
hány = how MANY – a specific number/unit/measurement of something . Can only be used with nouns.
mennyi = how MUCH (quantity) – less specific measurements, more about general amount than exact units or measurement (i.e. mennyi sört kérsz vs hány pohár sört kérsz?). Can only be used with nouns.
mennyire = how MUCH (to what extent) – used with adjectives or verbs
annyi = that much, so much/many (of that) – used mainly when talking about things that are distant and not immediately visible or present (same applies to annyira). Only used with nouns.
annyira = that much (to that extent) – used with adjectives or verbs
ennyi = this much, so much/many (of this) – used mainly when talking about things that are either present/visible at the time, or were recently mentioned (same applies to ennyire). Only used with nouns.
ennyire = this much (to this extent) – used with adjectives or verbs
Better Hungarian 
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