"Quasiperiodic" means that there is a function $L:\mathbb N\to\mathbb N$ such that any sequence of length $\ell$ appears in any sequence of length $L(\ell)$. You can get a quasiperiodic sequence as a limit, but you you will not get more. The later follows from your remark (which is added later).
P.S. If you start with a quasiperiodic sequence, then you can get a new sequence as a limit. But, the same way, you also can get your original sequence from the new one.
P.P.S. It seems to be interesting to consider quasiperiodic sequences with $L(\ell)=const\cdot\ell$. Was it considered anywhere? Is it true that if $L=\tfrac{3}{2}\cdot\ell$ then sequence is periodic?