Are there test statistics for which a larger sample size does not tend to produce a smaller p-value? I'm assuming that there is a genuine population effect.
By test statistics I mean test statistics that are in accepted use, and not a hypothetical test statistic that a reader might simply make up.
If the notion of a test statistic for which a larger sample size does not tend to produce a smaller p-value is incoherent, why?
 A: As said by Glen_b, the short answer is no as this would make the test inconsistent.
However, you can achieve what you want by using a compromise power analyses in which the critical values are changed as a function of $N$. Let me cite the relevant part from Faul et al. (2007, pp. 176), the paper which presents G*Power 3, the software for running such analyses:

In compromise power analyses (Erdfelder, 1984; Erdfelder et al., 1996;
  Müller, Manz, & Hoyer, 2002), both $\alpha$ and $1 - \beta$ are
  computed as functions of the effect size, $N$, and the error
  probability ratio $q = \beta/\alpha$. To illustrate, setting $q$ to 1
  would mean that the researcher prefers balanced Type I and Type II
  error risks ($\alpha = \beta$), whereas a $q$ of 4 would imply that
  $\beta = 4\alpha$ (cf. Cohen, 1988). Compromise power analyses can be
  useful both before and after data collection. For example, an a priori
  power analysis might result in a sample size that exceeds the
  available resources. In such a situation, a researcher could specify
  the maximum affordable sample size and, using a compromise power
  analysis, compute $\alpha$ and $1-\beta$ associated with, say, $q = \beta/\alpha=4$. Alternatively, if a study has already been conducted
  but has not yet been analyzed, a researcher could ask for a reasonable
  decision criterion that guarantees perfectly balanced error risks
  (i.e., $\alpha = \beta$) given the size of the sample and the critical
  effect size in which he or she is interested. Of course, compromise
  power analyses can easily result in unconventional significance levels
  greater than $\alpha = .05$ (in the case of small samples or effect
  sizes) or less than $\alpha = .001$ (in the case of large samples or
  effect sizes). However, we believe that the benefit of balanced Type I
  and Type II error risks often offsets the costs of violating
  significance level conventions (cf. Gigerenzer, Krauss, & Vitouch,
  2004).

References:
Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39(2), 175–191. http://doi.org/10.3758/BF03193146
