Datasets constructed for a purpose similar to that of Anscombe's quartet I've just come across Anscombe's quartet (four datasets that have almost indistinguishable descriptive statistics but look very different when plotted) and I am curious if there are other more or less well-known datasets that have been created to demonstrate the importance of certain aspects of statistical analyses.
 A: In the paper "Let's Put the Garbage-Can Regressions and Garbage-Can Probits Where They Belong" (C. Achen, 2004) the author creates a synthetic data set with a non-linearity that is meant to reflect real-life cases when data might have suffered a coding error during measurement (e.g. a distortion in assigning data to categorical values, or incorrect quantization procedures).
The synthetic data is created from a perfect linear relationship with two positive coefficients, but once you apply the non-linear coding error, standard regression techniques will produce a coefficient that is of the wrong sign and also statistically significant (and would become more so if you bootstrapped a larger synthetic data set).
Though it is just a small synthetic data set, the paper presents a great refutation of naive "dump everything I can think of on the right hand side" sorts of regression, showing that with even tiny / subtle non-linearities (which actually are quite common in things like coding errors or quantization errors), you can get wildly misleading results if you just trust the output of standard regression push-button analysis.
A: With regard to generating (e.g., your own) datasets for similar purposes, you might be interested in:


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*Chatterjee, S. & Firat, A. (2007). Generating data with identical statistics but dissimilar graphics: A follow up to the Anscombe dataset. The American Statistician, 61, 3, pp. 248–254.  


As far as datasets that are simply used to demonstrate tricky / counter-intuitive phenomena in statistics, there a lot, but you need to specify what phenomena you want to demonstrate.  For example, with respect to demonstrating Simpson's paradox, the Berkeley gender bias case dataset is very famous.  
For a great discussion of the most famous dataset of all, see: What aspects of the "Iris" data set make it so successful as an example/teaching/test data set.
