The way you have writen Gompetz, it does not get updated when you change the initial data. One way to solve the problem is to use With inside Manipulate, as in
Manipulate[
With[{sol =
NDSolveValue[{x'[t] == x[t], x[0] == x0}, x, {t, 0, 1}]},
Plot[sol[t], {t, 0, 1}, PlotRange -> {-E, E}]],
{x0, -1, 1}]
Alternatively, you can indicate explicitly the parameters when you define the solution, as in
sol[x1_] :=
sol[x1] = NDSolveValue[{x'[t] == x[t], x[0] == x1}, x, {t, 0, 1}];
Manipulate[
Plot[sol[x0][t], {t, 0, 1}, PlotRange -> {-E, E}],
{x0, -1, 1}]
so that sol[x1] does get updated when you change x1.