"Can conjoined triplets (or other higher multiples) exist? Dr. Rowena Spencer's 2003 book Conjoined Twins: Developmental Malformations and Clinical Implications discusses several cases of supposed conjoined triplets and quadruplets, most of which consist of one autosite with a number of parasites or acardiac twins (see above). However, a 2004 article in The American Journal of Obstetrics and Gynecology describes a case of parapagus dicephalus dibrachius dipus twins with triplet joined to the shared sternum, in the manner of xiphopagus twins. All three fetuses were well-formed and had approximately normal heads and extremities. This article provides conclusive proof that conjoined triplets, although extremely uncommon, can occur."
Only a tiny number of cases of conjoined triplets of quads has ever been reported - and of these, the third or fourth 'twin' was most usually discovered to be a deceased parasitic foetus.
There have also been very rare cases of triplet births which featured a set of conjoined twins - the most recent being that of Nida and Hira Jamal in Pakistan in 1992. The sisters were joined at the back, but had a completely healthy single sibling, Faryal.