What are some examples of psychology studies with tiny effect sizes and tiny p-values that are good for teaching?
One example might be the effect of birth order on intelligence. Given a large enough sample there highly significant effects (p&lt;.00001). However, these effects are very very small.

For a recent example, see Rohrer, Egloff, and Schmukle (2015): the effect of one birth order position was only one tenth of a standard deviation (1.5 IQ points)


  To illustrate this small effect, in our between-family sample of sibships of two, a randomly picked firstborn had a 52% chance of having a higher IQ than a randomly picked secondborn; conversely, a secondborn had a 48% chance of having a higher IQ than a firstborn. 


References

Rohrer, J. M., Egloff, B., &amp; Schmukle, S. C. (2015). Examining the effects of birth order on personality. Proceedings of the National Academy of Sciences, 201506451. http://doi.org/10.1073/pnas.1506451112