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$\begingroup$Have you seen the Mathematica function called Nest? Also NestList Would that solve your problem? Try NestList[Times[Exp[x], #] &, E0, 5]$\endgroup$
$\begingroup$Fold and FoldList could also be of interest here, in case the second argument $e^x$ changes from one step to the next. Because multiplying repeatedly by the same factor can be abbreviated by a power construct: $e^x\cdot e^x=e^{2x}$ etc. very generally, even if $x$ is a matrix.$\endgroup$
$\begingroup$Glad it helped, just a heads up: Your question may be put on-hold as it seems to be off-topic (see the help center), i.e it arises from a simple mistake and is unlikely to help any future visitors. In this case you where missing a function that is easily found in the documentation. Don't be discouraged by that cleaning-up policy. Your future good questions (see How to Ask) are welcome. Learn about common pitfalls. And do take the tour!$\endgroup$
Nest? AlsoNestListWould that solve your problem? TryNestList[Times[Exp[x], #] &, E0, 5]$\endgroup$FoldandFoldListcould also be of interest here, in case the second argument $e^x$ changes from one step to the next. Because multiplying repeatedly by the same factor can be abbreviated by a power construct: $e^x\cdot e^x=e^{2x}$ etc. very generally, even if $x$ is a matrix. $\endgroup$