In an earlier post, "Fixed Effects Without Panel Data", I argued that you could allow for (and indeed estimate) fixed effects in pure cross sections (i.e., no need for panel data) by using regularization estimators like LASSO. The idea is to fit a profligately-parameterized model but then to recover d.f. by regularization.
Note that you can use the same idea in time-series contexts. Even in a pure time series, you can allow for period-by-period time effects, broken polynomial trend with an arbitrary number of breakpoints, etc., via regularization. It turns out that a fascinating small literature on so-called "indicator saturation estimation" pursues this idea. The "indicators" are things like period-by-period time dummies, break-date location dummies, etc., and "saturation" refers to the profligate parameterization. Prominent contributors include David Hendry and Soren Johanssen; see this new paper and those that it cites. (Very cool application, by the way, to detecting historical volcanic eruptions.)