An infinite sequence of s and 1s, defined in any of the following (equivalent) ways. In each case, an obvious "limit" exists (and can be made rigorous), which is the Morse sequence

1. Write down a 0. Then, at each stage, copy the previous stage's word, then copy it "inverted", switching 0's and 1's.
2. Define a morphism by

   0 → 01
   1 → 10
   

   and extend it to words; now repeatedly apply it to the word "0".

In each case, the same sequence is obtained:

0
01
0110
01101001
0110100110010110
...

This sequence is aperiodic, but still strongly recurrent.