Jacobson's book on "Lie algebras" has the following definition of enveloping algebra generated by a subset (Definition 2, Chap II) : Start with an unital associative algebra $A$ (over a field $F$) and $S \subset A$. The enveloping algebra $S^{\ast}$ is simply the associative subalgebra of $A$ containing $1_A$ generated by $S$. The way I see it as $S^{\ast} = \sum_{n \geq 0} S^n$ where $S^n$ is the $F$-submodule generated by the set $\{ s_1 \dotsc s_n ~:~ s_1, \dotsc, s_n \in S \}$ of $n$-fold monomials (of course $S^{0} := F$). Since we require $1_A \in S^{\ast}$, we need $n=0$ in the sum.
Now while describing the subsequent properties, it says for $w \in A$ we have $\{ w \}^{\ast}$ is the algebra of polynomials in $w$ with constant term $0$. How can we have the constant terms $0$ if I need $1_A \in \{ w \}^{\ast}$?
You've misread Jacobson (p32 in his Dover book).
In a unital associative algebra $A$, he defines $S^*$ as the non-unital subalgebra generated by $S$ and calls it "enveloping associative algebra of $S$ in $A$", and $S^\dagger$ as the unital subalgebra generated by $S$ and calls it "enveloping algebra of $S$ in $A$". This is consistent with his description of $S^*$ as polynomials with 0 constant term.
(This is terrible terminology! But mathematicians from the XXth century built an irreversibly messy terminology with rings and algebras...)