  We show that the homotopy groups of a connective $E_k$-ring spectrum with an
$E_k$-cell attached along a class $\alpha$ in degree $n$ are isomorphic to the
homotopy groups of the cofiber of the self-map associated to $\alpha$ through
degree $2n$. Using this, we prove that the $2n-1^{st}$ homotopy groups of
Ravenel's $X(n)$ spectra are cyclic for all $n$. This further implies that,
after localizing at a prime, $X(n+1)$ is homotopically unique as the
$E_1$-$X(n)$-algebra with homotopy groups in degree $2n-1$ killed by an
$E_1$-cell. Lastly, we prove analogous theorems for a sequence of $E_k$-ring
Thom spectra, for each odd $k$, which are formally similar to Ravenel's $X(n)$
spectra and whose colimit is also $MU$.
