It has recently been established that quantum many-body scarring can prevent the thermalisation of some isolated quantum systems, starting from certain initial states. One of the first models to show this was the so-called PXP Hamiltonian, which was used to theoretically model an experiment on a chain of strongly interacting Rydberg atoms. A defining feature of the PXP Hamiltonian is a set of dynamical constraints that make certain states inaccessible to the dynamics. In this paper we construct a class of spin chain models that are parameterised by a discrete variable ℓ that controls the "strength" of a dynamical constraint. We show that by increasing ℓ the constraint becomes weaker, in the sense that fewer states are excluded from the dynamics. The PXP Hamiltonian is special case for ℓ=2. By weakening the constraint to ℓ≥4, however, we find a more extreme version of quantum scarring than in the PXP Hamiltonian, with the number of scar states growing exponentially in the system size.