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An alternative to @tolga's answer (which solves well this particular problem) is to define a recursive version of simplify_full. Very roughly :

sage: def rsf(x):
----:     "Recursive full_simplify-ication"
....:     ops=x.operands()
....:     if len(ops)==0: return x
....:     sops=list(map(rsf, ops))
....:     return x.operator()(*sops).simplify_full()
....: 
sage: rsf(exp(x/(x+1)+1/(x+1)))
e

No guarantee on performance, though...

Other alternatives :

sage: (exp(x/(x+1)+1/(x+1)))._sympy_().simplify()._sage_()
e
sage: giac.simplify(exp(x/(x+1)+1/(x+1))).sage()
e
sage: fricas.simplify(exp(x/(x+1)+1/(x+1))).sage()
e
sage: mathematica.FullSimplify(exp(x/(x+1)+1/(x+1))).sage()
e