  Globally coupled maps (GCMs) are prototypical examples of high-dimensional
dynamical systems. Interestingly, GCMs formed by an ensemble of weakly coupled
identical chaotic units generically exhibit a hyperchaotic 'turbulent' state. A
decade ago, Takeuchi et al. [Phys. Rev. Lett. 107, 124101 (2011)] theorized
that in turbulent GCMs the largest Lyapunov exponent (LE), $\lambda(N)$,
depends logarithmically on the system size $N$:
$\lambda_\infty-\lambda(N)\simeq c/\ln N$. We revisit the problem and analyze,
by means of analytical and numerical techniques, turbulent GCMs with positive
multipliers to show that there is a remarkable lack of universality, in
conflict with the previous prediction. In fact, we find a power-law scaling
$\lambda_\infty-\lambda(N)\simeq c/N^{\gamma}$, where $\gamma$ is a
parameter-dependent exponent in the range $0<\gamma\le1$. However, for strongly
dissimilar multipliers, the LE varies with $N$ in a slower fashion, which is
here numerically explored. Although our analysis is only valid for GCMs with
positive multipliers, it suggests that a universal convergence law for the LE
cannot be taken for granted in general GCMs.
