It is reported to have been sparked by an exploding fridge.
Witnesses described how they watched people jump out of the high-rise block, while a baby was dropped out of a window and caught on the street below.
Six hospitals across the capital are treating 78 people, with 18 in critical care.
Police warned the number of dead is expected to rise during "a long and complex recovery operation" but it was too early to say whether the death toll would increase considerably. They could not put a number on those missing.
Commander Stuart Cundy of the Metropolitan Police said: "Whilst we may have accessed every floor, that is not the same as a full search of the whole building.
"Sadly I don't anticipate that there will be further survivors."
Emergency teams have been seen carrying bodies from the remains of the building.
Grenfell Tower, built in 1974, contains 120 flats and is thought to have been home to between 400 and 600 people.
Friends and relatives of those missing are using social media to appeal for information about their loved ones.
More than 250 firefighters and 44 fire engines battled the blaze on Wednesday, with some sustaining "minor injuries".
A structural engineer assessed the tower's stability but judged the building was not in danger of collapsing.
Sky News Correspondent Mark White, who is on the Latimer Road side of the tower block, reported the fire has now "for the most part died down" but "every now and again there's still an eruption of flames from one of the window".
He added firefighters had been "making their way systematically through flat after flat, room after room" as they search through the debris and collapsed ceilings for those unaccounted for.
Steve Apter, director of safety and assurance at London Fire Brigade, revealed firefighters have got through "particularly arduous conditions" to reach the top floor of the tower.
Their work is expected to last into Thursday, with drones also being used to search the building.
The inferno affected every floor of the tower block, which is on the Lancaster West Estate, near the Westfield shopping centre.
London Fire Commissioner Dany Cotton described the blaze as an "unprecedented incident", something she had not seen in her 29 years of service.
She said the cause of the blaze was not yet known and was being investigated.
The fire has led to the closure of London Underground services between Hammersmith and Edgware Road on the Circle and Hammersmith and City lines.
The A40 is closed between Marylebone Road and the Northern Roundabout with other local roads and bus services affected.
As the fire ravaged the tower block, loud bangs could be heard as the blaze shattered windows in the tower, while debris and ash fell to the ground.
Mahad, who escaped Grenfell Tower, said the blaze began when his neighbour's fridge exploded.
He told Sky News: "There was no alarm, there was no bell, there was no sense of urgency.
"My neighbour is the one who knocked on our door. He said it was his house where the explosion happened. He said his fridge had exploded."
Other residents said Muslims who were awake to eat before beginning their daily Ramadan fast raised the alarm when the fire broke out.
A local residents group said its previous warnings about the tower's safety had fallen on "deaf ears".
In its blog post dated from November 2016, the group warned "only a catastrophic event" would expose the issues residents had with the building - including its single entry and exit point.
Labour MP Jim Fitzpatrick, secretary of the all-party Parliamentary Fire Safety and Rescue Group, said the Government has resisted calls to install sprinkler systems in high-rise blocks in the wake of the 2009 Lakanal House tragedy.
Grenfell Tower was recently refurbished at a cost of £8.6m - including the installation of new exterior cladding - with the work completed in May last year.
Following the blaze, the Government announced emergency checks are to be carried out on tower blocks refurbished in the same way.
Kensington and Chelsea Tenant Management Organisation, which manages the tower block, said it was too early to speculate on the cause of the fire.
It said: "We are aware that concerns have been raised historically by residents. We always take all concerns seriously and these will form part of our forthcoming investigations.
"While these investigations continue with our co-operation, our core priority at the moment is our residents."
Residents of the tower block have been encouraged to go to the nearby Westway Sports Centre as other relief centres reach capacity.
Sky News Senior Correspondent David Bowden, who is at one of the centres, reported a "very, very tense situation" as families wait to receive news of their loved ones.
He said: "It's all boiling over and police now are just trying to calm it down."
The Salvation Army and St Clements Church are urging the public to donate clothes, food and toiletries for those displaced by the fire.
London mayor Sadiq Khan said a "great many questions" would be asked about the fire, which has been declared a "major incident".
Labour leader Jeremy Corbyn suggested spending cuts to local authorities funding could have contributed to the blaze as he claimed "searching questions" would come from the tragedy.
Prime Minister Theresa May said there will be a "proper investigation", adding "If there are any lessons to be learned they will be, and action will be taken."
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