The government quickly said no after the video began spreading on social media Thursday.
A government statement yesterday said President Michelle Bachelet talked with Valentina Maureira and her father for more than an hour at the Catholic University hospital in the capital.
Officials did not release any information on what was said. But the government provided photographs of the visit, including one of Valentina taking a selfie with the president, who is also a pediatrician.
"I urgently request to speak to the president because I'm tired of living with this illness," said the teenager, whose older brother died at age 6 from the same disease.
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"I want her approval so I can get a shot that will make me sleep forever."
Cystic fibrosis damages multiple organs, especially the lungs, by causing recurrent infections that damage tissue.
Valentina said she was frustrated by the lack of options and by how the disease had hurt her quality of life.