Summary: We consider the chemotaxis system
\[ \begin{cases} u_t= \Delta u-\chi\nabla\cdot (u\nabla v)+g(u), &x\in\Omega,\;t>0,\\ 0= \Delta v-v+u, &x\in\Omega,\;t>0, \end{cases} \]
in a smooth bounded domain \(\Omega\subset\mathbb R^n\), where \(\chi>0\) and \(g\) generalizes the logistic function \(g(u)=Au-bu^\alpha\) with \(\alpha>1\), \(A\geq 0\) and \(b>0\). A concept of very weak solutions is introduced, and global existence of such solutions for any nonnegative initial data \(u_0\in L^1(\Omega)\) is proved under the assumption that \(\alpha>2-1/n\). Moreover, boundedness properties of the constructed solutions are studied. Inter alia, it is shown that if \(b\) is sufficiently large and \(u_0\in L^\infty(\Omega)\) has small norm in \(L^\gamma(\Omega)\) for some \(\gamma>n/2\) then the solution is globally bounded. Finally, in the case that additionally \(\alpha>n/2\) holds, a bounded set in \(L^\infty(\Omega)\) can be found which eventually attracts very weak solutions emanating from arbitrary \(L^1\) initial data. The paper closes with numerical experiments that illustrate some of the theoretically established results.