  At each prime $p$ and height $n+1 \ge 2$, we prove that the telescopic and
chromatic localizations of spectra differ. Specifically, for $\mathbb{Z}$
acting by Adams operations on $\mathrm{BP}\langle n \rangle$, we prove that the
$T(n+1)$-localized algebraic $K$-theory of $\mathrm{BP}\langle n
\rangle^{h\mathbb{Z}}$ is not $K(n+1)$-local. We also show that Galois
hyperdescent, $\mathbb{A}^1$-invariance, and nil-invariance fail for the
$K(n+1)$-localized algebraic $K$-theory of $K(n)$-local
$\mathbb{E}_{\infty}$-rings. In the case $n=1$ and $p \ge 7$ we make complete
computations of $T(2)_*\mathrm{K}(R)$, for $R$ certain finite Galois extensions
of the $K(1)$-local sphere. We show for $p\geq 5$ that the algebraic $K$-theory
of the $K(1)$-local sphere is asymptotically $L_2^{f}$-local.
