We contribute an answer to a quantitative variant of the question raised in [Coron, Contemp. Math 2007] concerning the controllability, in the framework of $L^2$ solutions, of the viscous Burgers equation $u_t+(u^2/2)_x=u_{xx}$ for initial and terminal data prescribed for $x\in(0,1)$. We investigate the (non)-controllability under the additional \emph{a priori} bound imposed on the (nonlinear) operator that associates the solution to the terminal state. In contrast to typical techniques on the controllability of the viscous Burgers equation invoking the heat equation, we combine scaling and compensated compactness arguments along with observations on the non-controllability of the inviscid Burgers equation to point out wide sets of terminal states non-attainable from zero initial data by solutions of restricted size. We prove in particular that, given $L\geq 1$, for sufficiently large $|C|$ and $T< (1+\Delta)/|C|$ (where $\Delta>0$ depends on $L$), the constant terminal state $u(\cdot,T):=C$ is not attainable at time $T$, starting from zero data, by weak solutions of the viscous Burgers equation satisfying a \textit{bounded amplification restriction} of the form $\|u\|_\infty\leq L|C|$. Further, in order to get closer to the original question we develop a basic well-posedness theory of unbounded entropy solutions to the Cauchy problem for multi-dimensional scalar conservation laws with pure $L^p$ data and polynomial growth up to the critical power $p$ of the flux function. The case of the Cauchy-Dirichlet problem for the Burgers equation on an interval is also addressed, in the $L^2$ solution framework which is considerably simpler than the established $L^1$ theory of renormalized solutions to such problems. Local regularity of unbounded entropy solutions is discussed in the one-dimensional case with convex flux. Non-controllability results are then extended to solutions of the viscous Burgers equation in the $L^2$ setting, under the bounded amplification restriction of the form $\|u\|_2\leq LT|C|$ and an additional $L^2-L^3_{loc}$ regularization assumption.