Suppose a test has ~$16.67\%$ power to detect some arbitrary but fixed effect size when sample size is $3$, and as we increase size by adding IID random observations to the sample ${4, 5, 6, 7,...}$ power approaches a limit ~ ${20.83\%, 21.67\%, 21.81\%, 21.83\%,...}$, so that more observations after a certain point provide no meaningful increase in information/precision? (In this example, power increases according to a convergent sequence, but that's just for convenience of illustration.) Is this possible?
Stating as a more general question: Can it be shown that for any random variable computed from an IID random sample of a population, and that contains information about a parameter of that population, we can always increase its information (precision) to any desired level (up to perfect information/exact precision/zero standard error) by including some number of additional IID random observations in the sample?
A couple of specifications: First, this could be a NHST statistic or an estimator--if the former, there's an upper limit on the probability of rejecting the null; if the latter, there's a lower limit on the standard error. Second, I'm referring to a property of the test (or statistic), not of the construct or the data collection method. Third, I'm not asking whether there are statistics with this property that people actually use. I assume no one would consider it practical. My question is about the theoretical possibility, and whether it's been discussed or proven (one way or the other) in the literature.
Fourth, I assume one could construct a trivial example of such a test by arbitrarily restricting how much information one uses from the sample as a function of sample size. For example, one might use a statistic that is the sum of all of the first sixteen observations, half of the next sixteen, a quarter of the next sixteen, and so on; or one might use all of the observations but intentionally add an amount of noise in proportion to sample size. I'm not necessarily interested in such examples, but if it can be shown that this is the only way to create such a statistic, or that no non-trivial examples of such a statistic have been found, that would be very interesting.