The Greek alphabet has ψ (psi) and ξ (ksi) , both letters correspond to a bi-consonantal sequence. Latin has X, Cyrillic alphabet has Щ which in some languages corresponds to ʃt͡ʃ, etc.
Normally, an alphabet is phonemic (before the language changes and departs from it). Is there any motivation for a single letter representing more than one consonant? how did such letters come to be?
In the case of Cyrillic Щ, it originated as just a scribal ligature of Ш and Т when writing Old Church Slavonic (where, naturally, it was pronounced /ʃtʲ/). It still does represent a cluster in some languages (e.g., Bulgarian /ʃt/, Ukrainian /ʃtʃ/), but in others has evolved into a single sound (e.g., [standard] Russian /ɕ:/).
The reason this particular letter stuck around is because it had a useful function: the cluster /ʃt'/ in Old Church Slavonic wasn't just any cluster, but it had an important morphophonemic function, being the jotated counterpart to /st/. In addition, Old Church Slavonic simply didn't have another good way of representing /ʃtʲ/, since palatalized /tʲ/ only occurred phonemically in this particular cluster; this probably encouraged more frequent use of the letter, although this is just me speculating.
I'm not sure, but I believe Greek ψ and ξ are similar, in that the clusters /ps/ and /ks/ have a special significance in the language, but I'm not very familiar with Greek.
Basically, morphophonemics can be a good motivator to have a single letter represent what's phonetically two sounds.
The Greek graphemes Ψ ('psi', [ps]) and Ξ ('xi', [ks]) represent consonant clusters in some varieties of the Greek alphabet. These clusters are represented by the same graphemes even when they are spilt by a morpheme boundary. Some sources describe these clusters as phonemes but this is not correct, they are combinations of two phonemes in all varieties of Greek.
It is not certain why the Greeks used single graphemes for these two clusters but it may be related to the fact that they are the only clusters that occur in coda) (ie. final) position in syllables. Earlier forms of the Attic alphabet lacked these graphemes and used ΦΣ and ΧΣ, respectively. While the majority of the Greek alphabet is believed to have been derived from either the Canaanite and/or Phoenician writing system, the graphemes Ψ and Ξ are have no obvious prototype in these traditions and may possibly have been derived from the South Arabian script.
While the alphabetic tradition normally has the target of representing individual phonemes, this may not be possible when borrowing an alphabet for use with a different language. Other considerations such as prosodic phenomena, syllable structure, morphophonemics, may make single graphemes for multiple phonemes a better solution than a straight mapping of segmental phonemes. And of course there are many languages which have single graphmes for phonemes, such as affricates and double articulations, that combine phones found in other languages as distinct phonemes.
In Greek, psi and ksi are used commonly in verb conjugations.
For instance, βάφω (I paint) becomes έβαψα (I painted), θα βάψω (I will paint) and βάψε (paint!). Similarly, τρέχω (I run) becomes έτρεξα (I ran), θα τρέξω (I will run) and τρέξε (run!).
I figure these two very common consonant clusters in so many verb conjugations lent itself to being written with one character. I believe there are no other consonant cluster so prevalent in verb conjugations.
