Ben van Beurden still sees fossil fuels as the company's future, despite concerns over climate change.
Trump has overturned Obama's ban and approved exploratory drilling, starting in December.
The Paris climate change conference on Monday must address US communities like Barrow, under assault from the weather.
Even the Mormon Church is profiting from land leases as low as $2 an acre, the Rainforest Action Network says.
"The Arctic is a unique treasure," Clinton tweeted. "Given what we know, it's not worth the risk of drilling."
The final permit came after the company repaired a ship essential to its exploration in the region this season.
The risk of another oil spill in Alaska is too high for BP to resume exploration.
Norwegian boss, who started in February, set to "move along" when takeover deal closes around early 2016.
Lawyers representing Nigerian farmers allege documents released by the UK courts show Shell lied in a previous case
Documents obtained by Amnesty show that the oil company ignored internal warnings that pipelines were in need of repair for six years before spills