The theory of classical independence and classical convolution can be generalised to noncommutative settings in several ways. The most famous one is that of free independence and free convolution (introduced by Voiculescu), but there is also boolean independence and boolean convolution (introduced by Speicher and Woroudi); monotone independence and monotone convolution (introduced by Muraki); and anti-monotone independence and anti-monotone convolution (the order-reversal of the previous notion). There are classification results of Speicher and Muraki that show that these are the only notions of independence (or convolution) that obey some natural set of axioms. (Speicher's classification assumed that convolution is commutative, so omitted the monotone and anti-monotone cases that were later discovered by Muraki.)
For each such concept of independence, there is a central limit theorem. Classically, the limiting distribution is the gaussian; in free probability it is the semicircular law; in the boolean case it is the Bernoulli distribution; and in the monotone and anti-monotone cases it is the arcsine law. See Section 9.2.1 of this recent thesis of David Jekel (and Chapter 5 of that thesis contains a more detailed history of the development of these notions of independence). For each of these independence concepts there is also an associated notion of entropy, and these distributions extremise the entropy amongst all distributions of a fixed mean and variance; again, Jekel's thesis has further information. (For the free case, of course, pretty much any introduction to free probability will contain these facts.)
EDIT: There is also finite free convolution, in which the analogue of the gaussian is the distribution of zeroes of Hermite polynomials.