  We prove new exponents for the energy version of the Erd\H{o}s-Szemer\'edi
sum-product conjecture, raised by Balog and Wooley. They match the previously
established milestone values for the standard formulation of the question, both
for general fields and the special case of real or complex numbers, and appear
to be the best ones attainable within the currently available technology.
Further results are obtained about multiplicative energies of additive shifts
and a strengthened energy version of the "few sums, many products" inequality
of Elekes and Ruzsa. The latter inequality enables us to obtain a minor
improvement of the state-of the art sum-product exponent over the reals due to
Konyagin and the second author, up to $\frac{4}{3}+\frac{1}{1509}$. An
application of energy estimates to an instance of arithmetic growth in prime
residue fields is presented.
