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I've often wondered if it would be feasible to have a searchable online database of worked out "first nontrivial examples". Users could upload their examples to the database and these could be community-refereed much like answers from this site or MathOverflow.

Certainly many textbooks contain such examples for undergrad material, but for research-level stuff this becomes more difficult to come by. It would be an efficient thing to crowdsource.

This question is a poor substitute for the above, but:

What are some hyperlinks to fully-worked "simplest nontrivial examples"?

In your answer, please indicate what your link will provide a basic nontrivial example of. If the example appears in a paper, please provide the page number and label for the example that is given in the paper.

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Similar questions have come up on MathOverflow. Here are some links.

https://mathoverflow.net/questions/16829/what-are-your-favorite-instructional-counterexamples has 46 answers.

https://mathoverflow.net/questions/29006/counterexamples-in-algebra has 39 answers.

https://mathoverflow.net/questions/68680/counterexamples-in-pde has 9 answers.

https://mathoverflow.net/questions/4994/fundamental-examples has 128 answers.

https://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism has 17 answers.

https://mathoverflow.net/questions/27357/what-are-your-favorite-finite-non-commutative-rings has 4 answers.

https://mathoverflow.net/questions/55365/counterexamples-in-algebraic-topology has 5 answers, but there are also a lot of examples in the question statement itself.

There are surely more, but that's a good start, I think.

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