  We study the extremes of variable speed branching Brownian motion (BBM) where
the time-dependent "speed functions", which describe the time-inhomogeneous
variance, converge to the identity function.
  We consider general speed functions lying strictly below their concave hull
and piecewise linear, concave speed functions.
  In the first case, the log-correction for the order of the maximum depends
only on the rate of convergence of the speed function near 0 and 1 and exhibits
a smooth interpolation between the correction in the i.i.d. case,
$\frac{1}{2\sqrt{2}} \ln t$, and that of standard BBM, $\frac{3}{2\sqrt{2}} \ln
t$.
  In the second case, we describe the order of the maximum in dependence of the
form of speed function and show that any log-correction larger than
$\frac{3}{2\sqrt{2}} \ln t$ can be obtained.
  In both cases, we prove that the limiting law of the maximum and the extremal
process essentially coincide with those of standard BBM, using a first and
second moment method which relies on the localisation of extremal particles.
  This extends the results of Bovier and Hartung for two-speed BBM.
