TITLE: Simplest nontrivial example repository
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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.