I will describe experiments performed on trapped atomic ions in which we examine various aspects of sequential quantum measurements. In the first, we perform measurements of modular functions of position and momentum on the oscillator state of a single ion. We measure correlations of such measurements, and explore conditions under which such measurements exhibit both no-signaling and signaling behaviors, experimentally comparing the latter as a test of quantum behavior with an alternative technique for violating a Leggett-Garg inequality. Measurements of modular position and momentum combined with preparation of squeezed states by reservoir engineering allow us to create and control approximate grid states, which have been suggested as candidates for fault-tolerant quantum error-correction with continuous variables. In a second line of work using a mixed-species chain containing both beryllium and calcium ions, we have demonstrated the repeated application of a quantum-non-demolition readout of the parity of qubits stored in the hyperfine states of two beryllium ions by mapping this information to a calcium qubit which is read out via state-dependent fluorescence. The use of two-species provides the opportunity to robustly recycle the measurement qubit, which is a key element for scaling up ion trap quantum computing. In a second setup, we work with sequential measurements on a qutrit stored in a calcium ion, which allows us to violate both state-independent and a state-dependent bounds on non-contextual hidden variable theories.