Vowel systems of different languages vary widely in size.
The ever-fascinating World Atlas of Language Structures Online devotes the second of its 142 “feature” articles to Vowel Quality Inventories. There is (i) a narrative discussion and (ii) a customizable map of the world plotting vowel systems with unusually small (2–4) or unusually large (7–14) inventories. Various theoretical or procedural issues, such as the status of diphthongs, have to be settled before you start counting. That done, though, the average number of vowels in a language turns out to be just fractionally below 6. The smallest vowel quality inventory recorded is 2 and the largest 14.

And which language would hold that record number of fourteen vowels? Standard German! (Count them: iː ɪ eː ɛ aː a yː ʏ øː œ uː ʊ oː ɔ, to which you could of course add ɛː for some speakers. The weak vowels ə ɐ are not counted, nor the nasalized vowels used in borrowed French words.) The “variety of British English included here” is reckoned to come in equal second place, with thirteen. Its posited inventory presumably comprises iː ɪ e æ ɑː ɒ ɔː ʊ uː ʌ ɜː eɪ əʊ, the last two being regarded as unitary. Not only the remaining diphthongs but also schwa are excluded.
Considerably more languages have an inventory of five vowels than any other number — just over a third of the sample. Familiar examples would be Spanish, Greek, Japanese and Swahili, all with just i e a o u.
Polish adds one more, ɨ, giving six vowels (ignoring the nasalized ones, which we can arguably analyse away). Korean adds two, ʌ ɯ, giving seven.
Four languages in the sample have only two contrasting vowel qualities.
You can see why many foreign learners of English (but not speakers of other Germanic languages) would find the English vowel contrasts so difficult to master. Not to mention foreign learners of German.
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