Hfrhyu

Besides the Latin script with its menagerie of diacritics and modified glyphs, what other phonetic scripts are extensible to such a degree to accommodate new sounds?

I know the Greek alphabet and Cyrillic shares many similarities with Latin in that diacritics and digraphs are employed to extend support for other phonologies.

I also know Japanese has some limited ability to represent voicing and vowel change (to represent foreign sounds like [fi]).

Korean has an extensive inventory of basic shapes which can be combined in various ways to cover new sounds (some extinct like ⟨ㆅ⟩ for [χ], and some hypothetical like ⟨ㅋ⟩ + ⟨ㅋ⟩ for [gʱ]).

While Latin is probably the most-extended script out there, many other writing systems have been extended in the same way.

Greek

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.

Coptic

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.

Cyrillic

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.)

Japanese kana

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.

Arabic

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

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.

Canadian Syllabics

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.

And more!

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!

Probably every alphabetic script is extensible in principle; more interesting is the question what alphabets with extensions are in practical use. To list a few

• Cyrillic has been extended in the Soviet Union to accommodate writing of Turkic languages and Caucasian languages. The main means of extension was the creation of new letter shapes.


• Arabic has been extended for Persian, Urdu, Sindhi, and a lot more languages


• Even the Greek alphabet has been extended historically, but the extensions for Coptic and Gothic are considered alphabets of their own right nowadays. On the other hand, the Baktrian letter Sho is still counted as a part of an extended Greek alphabet


• Devanagari has been extended with additional vowel signs for European vowels (mainly from English) and also with some additional consonants not present in Sanskrit but in modern languages of India (Both Indo-Aryan and Dravidian languages).