  We prove the existence and uniqueness of global solutions to the semilinear
stochastic heat equation on an unbounded spatial domain with forcing terms that
grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du = +\infty$
along with additional restrictions. For example, consider the forcing $f(u) = u
\log(e^e + |u|)\log(\log(e^e+|u|))$. A new dynamic weighting procedure is
introduced to control the solutions, which are unbounded in space.
