Gordon, Power, and Street proved that every tricategory (that is, weak 3-category) is equivalent to a -category. Not every tricategory is equivalent to a fully strict 3-category; any doubly-degenerate braided monoidal category which is not symmetric is an example. So this is “the best one can do” in one sense, although there are other incomparable paths one can take, such as weakening units but keeping interchange strict.
The inclusion of -categories into tricategories is not uniquely determined – there is a left and right-hand version (from a remark in Example 9.3.9 of Leinster’s book cited below). However, the two possible ways are canonically equivalent as tricategories.
A Gray-category with one object is called a Gray-monoid, and is a semi-strict version of a monoidal bicategory.
Since any tricategory is equivalent to a Gray-category, one can obtain examples of Gray-categories in this way. For example, the tricategory Bicat of bicategories, pseudofunctors, pseudonatural transformations, and modifications is equivalent to some Gray-category.
It is important to note that Bicat is not equivalent to Gray, due to the absence of pseudofunctors in the latter. It is equivalent to the sub-Gray-category of Gray determined by the “flexible” or “cofibrant” 2-categories, however, since between such 2-categories any pseudofunctor is equivalent to a strict one.
Since pseudofunctors between strict 2-categories compose strictly associatively, and between any 2-categories and there is a strict 2-category of pseudofunctors, pseudonatural transformations, and modifications, one might hope that there is a Gray-category consisting of strict 2-categories, pseudofunctors, pseudonatural transformations, and modifications, despite the fact that the prototypical example contains only strict 2-functors. However, this is false, because in a Gray-category the whiskering of 2-cells by a 1-cell is strictly functorial relative to composition of 2-cells along 1-cells, but this fails for whiskering of pseudonaturals by a pseudofunctor.
Gordon, Power, Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558, vi+81 pp.
Nick Gurski, Algebraic tricategories, Ph. D. Thesis.
Tom Leinster, Higher operads, higher categories