In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns themselves are uncorrelated with their own past if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares are not: there is time dependence in the variances, which cluster in time, with periods of high variance in volatile times.
Here is an artificial example (yet again, I know, but "real" stock return series may well look similar):
You see the high volatility cluster around in particular $t\approx400$.
Generated using
library(TSA)
garch01.sim <- garch.sim(alpha=c(.01,.55),beta=0.4,n=500)
plot(garch01.sim, type='l', ylab=expression(r[t]),xlab='t')