When did the first annular eclipse happen? A few hundreds of millions of years ago, the moon was closer to the Earth than it is today and hence was of a bigger apparent size. This made every solar eclipse either total or partial. Over the course of time, due to tidal forces between the moon and the Earth, the moon slowly drifted far from Earth decreasing its apparent size when viewed from Earth and hence causing annular eclipses to happen.

That would mean there must have been some day in history when Earth witnessed its first ever annular eclipse. Is there an estimate on when this happened?
A quick and dirty estimate is as follows.

An angular eclipse occurs first if the Moon is at apogee and Earth at perihelion. Assuming that these two events coincide (which happens every ~8.85 years owing to lunar apsidal precession), an angular eclipse requires that ($a$ denoting orbital semi-major axis and $R$ radius)

$$
      \frac{R_\supset}{(1+e_\supset)a_\supset} &lt; \frac{R_\odot}{(1-e_\oplus)a_\oplus}
$$

or

$$
      a_\supset &gt; \frac{R_\supset}{R_\odot} \frac{1-e_\oplus}{1+e_\supset} a_\oplus\approx 348300\,\mathsf{km}
$$

where I assumed that the orbital eccentricities, Lunar and Solar radii, and Earth's semi-major axis didn't change (over the period considered here) and are given by
$e_\supset=0.0549$, $e_\oplus=0.0167$, $R_\supset=1737.1\,$km, $R_\odot=695510\,$km, and $a_\oplus=149.6\times10^6\,$km.

At the present we have $a_\supset=384400\,$km. If we assume an average annual change by 22 mm, we find a time of 

$$1.6\times10^9\,\mathsf{years},$$

though it could have been more recent if the annual change of $a_\supset$ was larger some hundred million years ago (which may well have been the case).

In a similar way, one may also estimate when the last full Solar eclipse will occur. This time, the Moon is at perigee and Earth at apohelion, when

$$
      a_\supset &lt; \frac{R_\supset}{R_\odot} \frac{1+e_\supset}{1-e_\oplus} a_\oplus\approx 400800\,\mathsf{km}
$$

which last occurs in $430\times10^6$ or $745\times10^6$ years, depending on whether the current lunar recession rate of 38mm/yr or the long-term average of 22mm/yr is used.