Exoplanet atmospheric retrieval algorithms employ a Bayesian framework to quantify the uncertainties in model parameters based on the resulting posterior distribution. While in principle the resulting posterior is independent of the chosen Bayesian computation method, in practice there are differences due to the uncertainty introduced by the finite number of samples used to approximate the posterior. Additionally, sampler shortcomings (such as requiring more iterations than is feasible or rogue chains not converging) can introduce error in the posterior approximation. Here we compare the compute cost and accuracy of a variety of nested sampling and Markov chain Monte Carlo algorithms using retrievals on synthetic and real data. This research was supported by the NASA Fellowship Activity under NASA Grant 80NSSC20K0682, NASA Exoplanets Research Program grant NNX17AB62G, and NASA Astrophysics Data Analysis Program grant NNX16AL02G.