While Latin is probably the most-extended script out there, many other writing systems have been extended in the same way.
In the "oldest" form of the Greek alphabet (i.e. the oldest form we consider Greek rather than Phoenician), a number of letters were missing. Phi, chi, psi, and omega (Φ Χ Ψ Ω) were later inventions to better fit Greek phonology. Some dialects invented even more letters, which didn't survive into modern times: the Arcadians used a letter that looked like И to indicate a ts sound, while the Bactrians used a letter that looked like Þ to indicate a ʃ sound.
The Coptic alphabet started out as a variant of Greek, but quickly added some Demotic Egyptian letters for the sounds Greek lacked: Ϣ ʃ, Ϩ h, Ϫ c, and more.
The Cyrillic alphabet has been extended almost as much as the Latin one, since it's used all across the former Soviet Union and beyond. For just a few examples, the letters Ђ, Њ, Љ, and Ґ weren't in the earliest forms of modern Cyrillic, instead being added by individual languages that needed them. (If you're most familiar with Russian, certain other characters like Ѣ might seem like extensions—but these are actually older characters that Russian, Ukrainian, and other "big-name" Slavic languages have lost.)
The kana have been extended to write many indigenous Japanese languages, such as Ainu. Kana like セ゚ (tse) don't exist in Japanese: in this case, tse was created by adding a "voiceless plosive" mark to katakana se. Okinawan has many more "extra" kana, created by adding extra strokes or loops to standard hiragana, but these aren't represented in Unicode.
Like with Latin and Cyrillic, the Arabic writing system has been spread far and wide through conquest and trading. The Arabic language itself has an enormous number of dialects with different pronunciations, so sometimes new letters are created to represent these; other times, the innovations are for use in a non-Arabic language (like Persian or Swahili). For just a couple examples, گ (k with an extra stroke) was created for g, چ (dʒ with extra dots) for tʃ, and ڠ (ɣ with extra dots) for ŋ.
Devanagari was originally developed for Sanskrit, but now it's used for over a hundred languages across India. Many of these have sounds Sanskrit didn't, and use extra characters to express them: क़ (ka with a dot) for qa, ॻ (ga with an underline) for ɠa, and so on.
The syllabics were originally developed by James Evans for one particular dialect of Cree, but they caught on and spread like wildfire, with different Aboriginal groups modifying and adapting them for their own languages. New series of symbols were made for new consonants, such as ᕋ ra, ᕙ fa, ᖬ ða, and ᘔ za. But given the divergent evolution, the same symbol might be used very differently in different languages, or they might have created different new symbols for the same sound.
This isn't an exhaustive list, just the ones I could think of off the top of my head. Others should feel free to add to this answer with their own examples!