Is it possible to derive all the other boolean functions by taking other primitives different of $NAND$? I was reading the TECS book (The Elements of Computing Systems), in the book, we start to build the other logical gates with a single primitive logical gate, the $NAND$ gate. With it, we could easily make the $NOT$ gate, then the $AND$ gate and then the $OR$ gate.
With the $NOT$, $AND$ and $OR$ gates, we could express any truth table through cannonical representation.
The book has the table given below, I was wondering if we could do the same by taking any other boolean function as primitive, I'm quite sure that it's not possible to do with both constant $0$ nor constant $1$. What about the others?


Your question concerned arbitrary ($n$-ary) boolean functions, yet you seemed to have restricted your search to $0$-ary, $1$-ary and $2$-ary functions. A collection $C$ of boolean functions that can generate all the others by composition is called functionally complete, and when $C$ is a singleton $\{f\}$ we call such $f$ a Sheffer function. Now, a full characterization of functionally complete collections of boolean functions was given in "The Two-Valued Iterative Systems of Mathematical Logic" (Post 1941). As this book is hardly legible, though, I would heartily recommend the explanation of the main theorem in "Post's functional completeness theorem" instead. In "Composition and clones of Boolean functions", by Pöschel and Rosenberg (first chapter of Boolean Models and Methods in Mathematics, Computer Science, and Engineering), the exact number of $n$-ary Sheffer functions is determined: $2^{2^n - 2} - 2^{2^{n - 1} - 1}$. So, if you go beyond the $2$ binary functions NAND and NOR you should find $56$ functionally complete ternary functions, $16\,256$ functionally complete quaternary functions, and so on.
