Because we can easily invent as many small variations of Turing-complete models of computation as we like (see comments below the question), an answer to this question should try to concentrate on relevant (and Turing-complete) models, i.e. models that have either been investigated in illuminating non-trivial ways, or are important for better understanding of actually available computing resources.

I have been exposed in non-trivial ways to [tape based Turing machines][1], [register machines][2] and [pointer machines][3]. It seem like the [wikipedia article on abstract machines][4] is intended to give an overview for related Turing machine equivalent models, but in its current form it is mainly a collection of useful keywords and links.

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I'm currently looking for models and investigations related to machines limited to [write once read many (WORM) memory][5] for large amounts of data. None of the abstract machine models I found so far investigated these. Is it possible to create a model of such a machine that is equivalent to a Turing machine in the sense of the question above? (**Edit:** It looks like it was [proved recently][6] that [Wang B-machines][7] achieve this. I haven't read the paper yet.) This question seems to be both non-trivial and interesting to me, contrary to the comments below the question, which is the main reason why I wrote this answer.


  [1]: http://en.wikipedia.org/wiki/Turing_machine
  [2]: http://en.wikipedia.org/wiki/Register_machine
  [3]: http://en.wikipedia.org/wiki/Pointer_machine
  [4]: http://en.wikipedia.org/wiki/Abstract_machine
  [5]: https://cs.stackexchange.com/q/18939
  [6]: http://arxiv.org/abs/1304.0053
  [7]: http://en.wikipedia.org/wiki/Wang_B-machine