In season 1 we see

Drogon and the dragon's siblings getting hatched

And in season 8 we see

that Drogon has grown to a huge size.

How fast do they grow compared to other dragons?

Books

Official Answer

We simply do not know that. GRRM is deliberately vague about time, distances etc so we can't correctly figure out the rate on our own either. He merely at max gives us a vague idea by giving us a year and that's about it. And even the years are rare, most of the time he just doesn't discuss time so that fans couldn't find errors in his calculations. To Quote GRRM on distances and time:

[How big is Westeros? Is it the size of Europe, or even larger?]

I have deliberately tried to be vague about such things, so I don't have obsessive fans with rulers measuring distances on the map and telling me Ned couldn't get from X to Y in the time I say he did.

However, if you really must know, you can figure out the distances for yourself. The Wall is a hundred leagues long. A league is three miles. Go from there.

But if you turn up any mistakes in travel times by using that measure, let it be your secret.
_{Citadel - So Spake Martin}

He also does not want people to apply real world physics, biology, chemistry etc to his magical world.

I have people constantly writing me with science fiction theories about the seasons — “It’s a double star system with a black dwarf and that would explain–” It’s fantasy, man, it’s magic.
_{A Dance with Dragons Interview with GRRM - EW}

Speculation

Let's take Joffrey Velaryon as the measuring stick here.

He was born in 117 AC at Dragonstone and a dragon egg was placed in his crib by command of his Royal Grandfather Viserys I. The dragon, named Tyraxes, hatched at some unknown point of time but it is estimated it was no later than 120 AC.

The Dance of the Dragon began in 129 AC, some 9 years after it, if we assume Tryaxes hatched in 120 AC. Bear in mind, Tyraxes was at Dragonstone, the place which was considered best for growth and health of dragons. Yet, after growing for 9 years, Tyraxes was still not old enough to fly to long distances alone
¹ but it was large enough to carry the young Prince. It was however considered too young for war, unlike the dragons Arrax and Vermax which were bonded to Joffrey's older brothers. We do not know if Tyraxes' slower growth was because it hatched late or simply because Joffrey's brothers' dragons were more robust.

We do know from Daenerys' dragons that three dragons born at the same time can have different growth rate e.g. Drogon always ate more, was more daring and grew faster and larger. Rhaegel and Viserion were not that physically robust and certainly their captivity in Meereen didn't help.

In any case, I believe since Drogon had started flying off to long distances and by the end of ADWD, he was large enough to carry Daenerys to the skies, he must have been slightly larger or at least the same size as Tyraxes was when the war began. Drogon, we do know, was born somewhere in 299 AC. We know Daenerys took Meereen within the same year. Tyraxes took 9 years to grow to that size, that too in the environment considered best for dragons. Drogon seems to have grown nearly that much, if not more within one to two years given that Daenerys' flight happens in Meereen, after second siege of Astapor which happened in 300 AC.

But it is possible that Tyraxes is the wrong measuring tape here, we perhaps should have used Arrax or Vermax for comparison with Drogon since Arrax and Vermax showed faster growth despite being born at more or less the same time as Tyraxes. And not to mention another flawed premise in our calculation, while Tyraxes could take Prince Joffrey to skies after 9 years, that doesn't necessarily mean that the first flight occurred after 9 years. So when did Prince Joffrey first flew atop his dragon? So now let's look at Joffrey's half brother Aegon the Younger (Later Aegon III). Aegon was born in 120 AC and just like his brothers, an egg was placed in his crib, which hatched into a Dragon named Stormcloud. When the war broke out 9 years later, the Prince had bonded with the dragon but hadn't flown since the Dragon was too small for him. The Prince's solo flight with his dragon was when he desperately clutch to his bleeding dragon's neck in order to escape from the attacking fleets of the Kingdom of Three Daughters. So after 9 years, Stormcloud still wasn't old enough to serve as a proper mount for Aegon III. But OTOH, we do not know when Stormcloud was hatched.

In any case, even if the dragons Tyraxes and Stormcloud weren't born in 120 AC and we are off by a few years, one thing seems clear, Daenerys' dragons have grown faster than other dragons.

_{1. Not deemed suitable to fly alone without a rider seems more of a comment on fear of losing a dragon rather than its inability to fly as such. A riderless young dragon presumably seeks out new lairs just as Drogon did with the hill in Dothraki sea that Dany dubbed her Dragonstone. Older dragons, I suppose, can be trusted to return back to the lair they are used to even if they are riderless i.e. Dragonstone or King's Landing. Dragons are solitary animals and they do not have a herd behaviour. In the Wild, Dragons live alone, presumably seeking company only to mate.}