For the purposes of this question I would interpret the word "recorded" as loosely as possible meaning that historic records such as ice cores or tree rings would count. Also what effect did such a large flare have on life on Earth at the time?
A spike in Carbon-14 content was recorded from tree rings in Japan, possibly due to a solar flare 20 times more energetic than the 1859 Carrington Event (the largest definitively recorded solar storm). This event has been dated to 774 AD. I think this could be the most energetic (tentatively) recorded solar flare.
Such a solar storm would be far more devastating today than it was back then, in the sense that we rely so much on technology that the EM interference would cost trillions of dollars in damage to power grids and communications.
A flare would need to be disastrously large, I think, to give us on Earth radiation poisoning. Certainly, animals that use Earth's magnetic field for navigation and would get rather confused. But our atmosphere and magnetic field are adequate protection when we're on Earth.
Here is a good write-up of the situation, with a couple of Nature papers if you have access.
If you're going to delve into considering the effects of solar storms on human health (i.e. the effects of magnetic field perturbations), put your tinfoil hat on first. There's a lot of pseudoscience and little substantive research.
What is the largest recorded solar flare?
Since @Moriarty already dealt with historical records, I will focus on the modern observations.
On April 2, 2001 the GOES spacecraft recorded an X20 flare (see the following for flare classification: https://en.wikipedia.org/wiki/Solar_flare#Classification). Most flares fall below C-class (i.e., at least 100 times weaker) and X-class flares are the strongest.
In events that also release coronal mass ejections (CMEs), we can classify the strength of the solar storm by the CME speed.
The 1859 Carrington event reached Earth in just ~17.67 hr, which is an average speed of ~2400 km/s (or ~5.3 million mph).
The record CME speed was set in August 1972, where the CME speed was estimated at ~2850 km/s (or ~6.4 million mph).
Probability of Occurrence
There was a recent study that found the probability of another Carrington-like event is as high as ~12% in the next decade.