Motivated by work of Coxeter (1957), we study a class of algebras associated to Coxeter groups, which we term 'generalized nil-Coxeter algebras'. We construct the first finite-dimensional examples other than usual nil-Coxeter algebras; these form a $2$-parameter type $A$ family that we term $NC_A(n,d)$. We explore the combinatorial properties of these algebras, including the Coxeter word basis, length function, maximal words, and their connection to Khovanov's categorification of the Weyl algebra. Our broader motivation arises from complex reflection groups and the Broue-Malle-Rouquier freeness conjecture (1998). With generic Hecke algebras over real and complex groups in mind, we show that the 'first' finite-dimensional examples $NC_A(n,d)$ are in fact the only ones, outside of the usual nil-Coxeter algebras. The proofs use a diagrammatic calculus akin to crystal theory.