Sequences containing all finite sequences

[hv at crypt.org]

franktaw at netscape.net (Franklin T. Adams-Watters) wrote:
:Which sequences contain all finite sequences of non-negative integers as subsequences?  (One can also add 1, and look for sequences containing just all finite sequences of positive integers as subsequences.  However, the natural examples I found all involved non-negative integers, so that's the way I'm framing the problem.)

This is a nice concept.

One sequence that maps more readily to the positive integers is the
concatenated list of all compositions of successive n:

1,
1,1, 2,
1,1,1, 1,2, 2,1, 3,
1,1,1,1, 1,1,2, 1,2,1, 1,3, 2,1,1, 2,2, 3,1, 4,
etc (not in OEIS, but cf A045623)

:Are there other sequences with this property in the OEIS - or that should be in the OEIS?  Note that all 3 of these are tables, where every finite sequence occurs in a row in an obvious way.  It would be nice to find sequences not defined in this way.

I'd expect all such sequences to grow very slowly, which may reduce their
usefulness within the OEIS other than when viewed particularly as examples
of this property.

It looks like A066099 may also be a positive integer example, though
possibly just a minor variation of your factorisation example;
I stumbled across it checking for presence of the above.

Hugo