This answer expands on Paul McKenney's comment, and concerns only the gap structure in $\mathcal{P}(\omega)$/fin. I don't know much about what happens for larger cardinals.
The fact that $\mathcal{P}(\omega)$/fin is highly incomplete actually has interesting consequences, and in particular is useful in showing that $\mathcal{P}(\omega)$/fin is universal for Boolean algebras of size at most $\aleph_1$. To give a better sense of the story here, suppose $\langle A_i: i < \omega \rangle$ is a strictly $\subseteq_*$-increasing sequence of subsets of $\omega$. Then if we iterate Paul's construction, we can find a strictly decreasing sequence $\langle B_j: j <\omega \rangle$ of upper bounds for $\langle A_i: i < \omega \rangle$. A natural question is whether the resulting pair of sequences $A_0 \subseteq_* A_1 \subseteq_* \ldots \subseteq_* B_1 \subseteq_* B_0$ is ever a gap. It turns out the answer is no: one can always find a $C$ such that for all $i, j$ we have $A_i \subseteq_* C \subseteq_* B_j$.
This says that there are no $(\omega, \omega)$-gaps in $\mathcal{P}(\omega)$/fin. Rothberger proved that the minimum cardinal $\kappa$ for which there exists an $(\omega, \kappa)$-gap is the bounding number $\mathfrak{b}$. If the CH holds we have $\mathfrak{b} = \omega_1$, and so in this case we can find $(\omega, \omega_1)$-gaps in as well as the more celebrated $(\omega_1, \omega_1)$-gaps of Hausdorff.
Even without CH, we have that $\mathcal{P}(\omega)$/fin has the following four properties:
- it is atomless (as a Boolean algebra)
- it has no countable limit points
- it has no countable cofinal sequences
- it has no $(\omega, \omega)$-gaps
One can prove from these that $\mathcal{P}(\omega)$/fin embeds every Boolean algebra of size at most $\aleph_1$. In the case of CH, any other Boolean algebra of size $\aleph_1$ with these four properties is isomorphic to $\mathcal{P}(\omega)$/fin. The proof for both facts goes by extending countable partial embeddings one element at a time. If there existed countable limit points in $\mathcal{P}(\omega)$/fin, as you asked about in your question, such a proof would not be possible: such points would block certain partial embeddings from being extended.
An encyclopedic reference for these results and the set theory of $\mathcal{P}(\omega)$/fin more generally is the book Hausdorff Gaps and Limits by Frankiewicz and Zbierski.