Discrete-time affine processes are widely used in finance and economics and encompass count, positive, and nonnegative-valued processes. This paper develops near-unit-root asymptotic theory for this class of models. Unlike linear AR(1) processes, affine processes exhibit time-varying conditional variance that remains asymptotically non-negligible near unity, leading to qualitatively different scaling limits and estimator behavior. We show that the local-to-unity regime suffers from the usual nuisance-parameter problem, whereas the mildly explosive regime, while free of it, still does not allow consistent estimation of the intercept. By contrast, the mildly stationary framework is more tractable: the OLS estimator is asymptotically normal, the resulting trajectories are more realistic than those of linear AR(1) models, and inference is possible through both a plug-in method or bootstrap. The theoretical results are supported by simulation evidence and illustrated through applications to insurance and financial data.