zlacker

You're right in focusing on the diversity of the parts. After all, a large pile of sand has a lot of parts, but they are all (to a decent approximation) the same. Therefore (?) they have the same small set of interactions, and the whole system has a 'small' set of states. (However, see 'self-organized criticality').

Software is where the diversity is, for computers. A cells systems are a set of interlocking networks of DNA, RNA, protein, small molecules, and signalling. As you say, these diverse parts then interact in diverse ways.

Another aspect is the large number of 'levels' in biological systems - humans have around seven; atoms, molecules, organelles, cells, tissues, organs, whole system. The interactions within and between these levels is a further source of complexity.