I thought it would be fun to look and see who the last person alive to see a living dragon might be. We know the dragons died out during the reign of Aegon III (The Dragonbane) who ruled from 131AC to 157AC (with the last dragon dying in 153AC).

If interested: Post-Dance Dragons: Last Sighting For Each & The Blood of Old Valyria III: The Last of the Dragons

Note: While there may be some dragons "in Asshai", etc. this post is regarding the known dragons in the series. If interested: Pre-Targaryen Dragons in Westeros

Potential Options

Ser Arlan of Pennytree (209AC)

Ser Arlan dies in 209AC, but saw the last dragon as a child in 152AC when he was quite young:

Not that Dunk had ever seen a dragon. The old man had, though. Dunk had heard the story half a hundred times, how Ser Arlan had been just a little boy when his grandfather had taken him to King's Landing, and how they'd seen the last dragon there the year before it died. She'd been a green female, small and stunted, her wings withered. None of her eggs had ever hatched. "Some say King Aegon poisoned her," the old man would tell. -The Hedge Knight

Elaena Targaryen (in or after 220AC)

Outliving her older sisters (Daena/Rhaena) Elaena may have seen a dragon (even though her egg never hatched) but she was born in 150 so she would have been quite young.

She lives to at least 220AC though:

Elaena outlived her siblings and led a tumultuous life once freed from the Maidenvault. Following in Daena's footsteps, she bore the bastard twins Jon and Jeyne Waters to Alyn Velaryon, Lord Oakenfist. She hoped to wed him, it is written, but a year after his disappearance at sea, she gave up hope and agreed to marry elsewhere.

and:

men said she was more beautiful at seventy than she had been at seventeen.-SSM, Three Maidens in the Tower

Worth noting that since it is confirmed that Elaena ever saw a dagon, her sisters Daena/Rhaena both have unconfirmed death dates after 171AC.

Laena Velaryon

Laena was given a dragon egg at birth that turned out to be a monstrosity:

The first omen of the dark times to come was seen on Driftmark, when the dragon’s egg presented to Laena Velaryon upon her birth quickened and hatched. Her parents’ pride and pleasure quickly turned to ash, however; the dragon that wriggled from the egg was a monstrosity, a wingless wyrm, maggot-white and blind. Within moments of hatching, the creature turned upon the babe in her cradle and tore a bloody chunk from her arm. As Laena shrieked, Lord Oakenfist ripped the “dragon” off her, flung it to the floor, and hacked it into pieces.

If interested: Targaryen "Monstrosities": Infants & Dragons

She was born in 134AC but has an unknown fate, but obviously would have seen other dragons in her life.

Note: This also doesn't count the numerous different attempts made over the years to try and bring these dragons back or as mentioned any possible dragons elsewhere in the world.

TLDR: Trying to figure out who the last person alive to see a dragon was (before Dany's hatching). If the last dragon died in 153AC, Dunk's mentor Ser Arlan might have been the last named character to see one alive. There are several other characters who might surpass him but none of them are confirmed due to the fact that Elaena would have been quite young (but probably saw one) and her sisters/Laena have unconfirmed deaths that could technically be after Ser Arlan.