Councils provided an extra 90,000 places last year, says the LGA, but it says another 130,000 will be needed in the next three years, as the birth rate continues to climb.
The pressure on space has meant that one school had to build its playground on a roof, says the LGA.
The LGA's survey, based on responses from over half of councils, found that three out of four local authorities claim not to have received sufficient funding from central government for extra places - and that 38% of councils have had to borrow.
The LGA says its research "lays bare the scale of the problem in funding for school places".
Councils have called for more control over the creation of extra places, complaining that the push for more autonomous academies and free schools has weakened strategic planning.
They also want a clearer funding mechanism for creating places.
"Councils are delivering, but at a cost. Since the pressure on places first emerged, councils have been getting on with the job of creating more, and welcome though government funding is, it is nothing like the full cost," said David Simmonds, chairman of the LGA's Children and Young People Board.
"The scale of this black hole is such that the cost of the creation of new school places cannot be met by council tax payers."
Mr Simmonds also warned that the pressure on places was moving up to secondary level and that another 200,000 places will be needed.