  This paper deals with the quasilinear parabolic-elliptic chemotaxis system
with logistic source and nonlinear production, \begin{equation*}
  \begin{cases}
  u_t=\nabla \cdot (D(u) \nabla u) - \nabla \cdot (S(u)\nabla v)
  + \lambda u - \mu u^{\kappa},
  & x\in\Omega,\ t>0,
  \\[1mm]
  0=\Delta v - \overline{M_f}(t) + f(u),
  & x\in\Omega,\ t>0,
  \end{cases} \end{equation*} where $\lambda>0$, $\mu>0$, $\kappa>1$ and
$\overline{M_f}(t):=\frac{1}{|\Omega|}\int_{\Omega} f(u(x,t))\,dx$, and $D$,
$S$ and $f$ are functions generalizing the prototypes \begin{align*}
  D(u)=(u+1)^{m-1},\quad
  S(u)=u(u+1)^{\alpha-1}\quad\mbox{and}\quad
  f(u)=u^\ell \end{align*} with $m\in\mathbb{R}$, $\alpha>0$ and $\ell>0$. In
the case $m=\alpha=\ell=1$, Fuest (NoDEA Nonlinear Differential Equations
Appl.; 2021; 28; 16) obtained conditions for $\kappa$ such that solutions blow
up in finite time. However, in the above system boundedness and finite-time
blow-up of solutions have been not yet established. This paper gives
boundedness and finite-time blow-up under some conditions for $m$, $\alpha$,
$\kappa$ and $\ell$.
