  We identify the simplest combinations of entanglement and incompatibility
giving the maximum quantum violation for each of the 46 classes of tight Bell
inequalities for the (3,2,2) scenario, i.e., three parties, two measurements
per party, and two outcomes per measurement. This allows us to classify the
maximum quantum nonlocality according to the simplest resources needed to
achieve it. We show that entanglement and incompatibility only produce maximum
nonlocality when they are combined in specific ways. For each entanglement
class there is, in most cases, just one incompatibility class leading to
maximum nonlocality. We also identify two interesting cases. We show that the
maximum quantum violation of \'Sliwa inequality 23 only occurs when the third
party measures the identity, so nonlocality cannot increase when we add a third
party to the bipartite case. Almost quantum correlations predict that adding a
new party increases nonlocality. This points out that either almost quantum
correlations violate a fundamental principle or that there is a form of
tripartite entanglement which quantum theory cannot account for. The other
interesting case is the maximum quantum violation of \'Sliwa inequality 26,
which, like the Mermin inequality, requires maximum incompatibility for all
parties. In contrast, it requires a specific entangled state which has the same
tripartite negativity as the W state.
