Example of a consistent estimator that doesn't grow less variable with increased sample size? I've had it asserted to me that any consistent estimator must necessarily also grow less variable with increased sample size.
I felt that this couldn't be correct, since there was nothing in the definition of a consistent estimator that forced this to be so, and the class of possible estimators is infinite.
What's an example of a consistent estimator that doesn't grow less variable with increased sample size?
I'd prefer the example be an estimator that's commonly used, rather than one contrived just to meet the demands of the question.
 A: The common meaning of "consistency" and its technical meaning are different. See this page for some discussion. Also, as noted by @hejseb in a comment on another answer here, lack of bias and consistency are not the same.
This quote from the Wikipedia page may help remove some confusion:

Bias is related to consistency as follows: a sequence of estimators is consistent if and only if it converges to a value and the bias converges to zero. Consistent estimators are convergent and asymptotically unbiased (hence converge to the correct value): individual estimators in the sequence may be biased, but the overall sequence still consistent, if the bias converges to zero. Conversely, if the sequence does not converge to a value, then it is not consistent, regardless of whether the estimators in the sequence are biased or not.

The estimator for the mean of a sequence proposed in another answer here:
$$X_1 + \frac{1}{n}$$
thus is not consistent because it does not converge to the true value of the mean as the number of observations increases.
The requirement for convergence means that the estimator must get arbitrarily close to to the true value as sample size increases. That would seem to require that the estimator "grow less variable with increased sample size," for any reasonable definition of "less variable."
