In Classical times, few digraphs were used.
The letters K, Y, and Z were used for Greek words, along with the digraphs Ph, Th, Ch for aspirated stops, but other words were generally assimilated to Latin phonology. (Source: searching for the combinations below in the Packhum corpus and finding nothing.)
Z was used for more foreign sounds in Late Latin.
Jordanes' De Origine Actibusque Getarum uses Z when transcribing the sounds /ts/ and /dz/ in foreign languages, as well as on its own for /z/:
- Tz: Emnetzur, Ultzindur, Dintzic, etc
- Dz: Scandza, Augandzi, Tadzans, etc
- Z: Boz, Bizzim
Other foreign sounds were still assimilated, but these particular combinations weren't utterly impossible in Latin (e.g. the compound etsi).
This tradition seems to have gotten slowly broader and broader over the next several centuries, as people with native Slavic names were mentioned in Latin texts (e.g. the 12th-century Bull of Gniezno with rz for a retroflex sibilant, ch for some sort of affricate, and z for various other Polish consonants; the 12th-century Chronica Slavorum with zc for /tʃ/).
Unfortunately, my sources are sparse, and more data from Classical or Late Latin would be appreciated: surely other writers than Jordanes mentioned foreign names.