How to identify if a variable is quasiperiodic? At the core of this question is an ignorance of what a quasiperiodic variable (or collection of variables in a multivariable system) actually is. I don't have a particular data set in mind, but rather I am seeking to expand my repertoire of patterns to look for in future exploratory data analysis of time series.
Wikipedia gives this description:

Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpredictability that does not lend itself to precise measurement. It is different from the mathematical concept of an almost periodic function, which has increasing regularity over multiple periods. The mathematical definition of quasiperiodic function is a completely different concept; the two should not be confused.

Their description tells me a few things about what a quasiperiodic variable isn't, but I still feel quite in the dark about what it is. What is it?

A quasiperiodic function is comprised of signals having no (pairwise) common periods.
Imagine, if you will, a lake in which one wave is traveling in one direction, another wave is traveling in a different direction, both waves were suddenly frozen into place, and you are dashing across the ice on skates at a constant speed.  Unless the waves have the same wavelengths, you will feel an irregular bounce as you travel even though the surface has such an obviously periodic (wavelike) structure.
In this plan view, one wave was traveling left to right and the other was traveling top to bottom with a wavelength $\tau = (1 + \sqrt{5})/2$ times the other.  The white line tracks your path.

The next figure plots the elevation of the surface versus time (from bottom left to upper right in the previous figure).

In the background I have painted red strips for every other left-to-right period and blue strips for every other top-to-bottom period.  The graph exhibits an intriguing "sort of periodic" behavior, but it's not perfectly periodic.  For instance, look at the tallest finger-like peaks: they are not regularly spaced.  But they will keep recurring.
All quasiperiodic functions arise this way (but may have more than two wavelike components and even a countable infinity of them).  When the wavelengths are incommensurable (none is a rational multiple of another), the graph never repeats.  Because of this, it is possible -- and often happens -- that when you wait long enough, the behavior of the graph appears to have completely changed.  There may be places along the frozen lake where just by coincidence you are skating over nearly horizontal ice and others where it's impassable.
The possibility of explaining an apparently chaotic, aperiodic signal in terms of a small number of perfectly periodic signals leads to powerful analyses.  I believe this form of analysis has been successfully performed for spatio-temporal modeling of large-scale meteorological variables (such as air temperatures and pressures over the oceans).

An almost periodic function is exactly that: when you translate its graph by a certain amount (its "almost period"), it almost stays the same.

Again I have shaded every other "almost period" in gray for reference.  The original graph is in light blue with a black border.  When translated by the almost period (shown by the gray shadow) it changes, but only a little.
There are different definitions of almost periodic functions, describing slightly different behaviors.  They vary according to how they measure the closeness of the function to its translate.  But in all cases, once you have observed the function for a whole period, it is likely to behave similarly for the next period, and the next one after that, and so on, changing only slowly as you go on.  Thus the next period of the series is reliably, but not perfectly, predictable from the current period.
With a decent time series analysis you might compare the data to their translations by one or more multiples of an (apparent) period and analyze the differences into something you can model plus random residuals.  When the signal is almost periodic, those differences will be a time series of substantially smaller amplitude than the original.  This sort of analysis underlies most seasonal time series decompositions, where it is hoped that (say) next year's values will behave much like this year's values, subject to relatively small long-term changes plus small random errors.
