Standard error is sigma / sqrt(n), so halving your uncertainty requires four times the data. That sqrt(n) law is the most useful practical consequence and it recurs everywhere from Sharpe estimation to Monte Carlo.
The condition people forget: finite variance. Cauchy-distributed samples never converge - the mean of n of them is distributed exactly like a single one. Naming this unprompted is a strong signal.
The finance caveat. Convergence is slowest in the tails, which is exactly where risk management operates. "Returns are normal by the CLT" is defensible for aggregate averages and dangerous for tail risk.