This was asked in Google Interview: https://www.glassdoor.com/Interview/Explain-a-probability-distribution-that-is-not-normal-and-how-to-apply-that-QTN_1812014.htm
Some examples: http://www.statisticshowto.com/probability-and-statistics/non-normal-distributions/
 

Common Probability distributions cheatsheet: https://blog.cloudera.com/blog/2015/12/common-probability-distributions-the-data-scientists-crib-sheet/

Give examples of data that does not have a Gaussian distribution, nor log-normal.

• Allocation of wealth among individuals
• Values of oil reserves among oil fields (many small ones, a small number of large ones)