We can interpret $\mathbb{R}[[X]]$ as the set of one sided sequences by interpreting $(a_0,a_1,\dots)$ as $a_0x^0+a_1x^1+\dots=\sum_{i=0}^{\infty}a_nx^n$.
Can we extend this to a field of two-sided sequences $(\dots,a_{-2},a_{-1},a_0,a_1,a_2,\dots)$ while preserving the ring structure of one-sided sequences and having termwise addition? That is: I want
$$(\dots,a_{-2},a_{-1},a_0,a_1,a_2,\dots)+(\dots,b_{-2},b_{-1},b_0,b_1,b_2,\dots)=(\dots,a_{-2}+b_{-2},a_{-1}+b_{-2},a_0+b_0,a_1+b_1,a_2+b_2,\dots)$$
and, if $a_i=0$ for all $i<0$, then I want the multiplication to be defined as in the formal power series ring.
Remarks: If one does not care about including all two-sided sequences, we can take the Laurant series.
If we want to include all two-sided sequences but we don't care about termwise addition, even if we want to preserve the ring structure where all terms with negative indices are 0, this is easy by finding a bijection from two-sided sequences to one-sided sequences sending any element that is already a one-sided sequence (i.e. all terms with negative indices are 0) to itself, and sending the rest to other Laurant series (existence is given by Schroder Bernstein)
Similarly, if we care about termwise addition but not inverses, we can use the bijection sending $a_i$ to $a_{2i}$ if $i\ge0$, and $a_i$ to $a_{-2i-1}$ if $i<0$ and use polynomial/power series multiplication.
Yes, this is possible, and you can additionally require the natural $\mathbb{R}$-vector space structure to be preserved. Consider the set $S$ of two-sided sequences as a vector space over $\mathbb{R}$ in the obvious way, with $\mathbb{R}[[X]]$ as a subspace. Let $C\subset S$ be the subspace of sequences with $a_i=0$ for all $i\geq 0$, so $S=\mathbb{R}[[X]]\oplus C$. Note that $C$ has dimension $2^{\aleph_0}$.
Now let $F$ be any field extension of $\mathbb{R}((X))$ such that the $\mathbb{R}$-dimension of the vector space $F/\mathbb{R}[[X]]$ is $2^{\aleph_0}$. (In fact, any nontrivial field extension of $\mathbb{R}((X))$ of cardinality $2^{\aleph_0}$ will have this property.) Let $D$ be a linear complement of $\mathbb{R}[[X]]$ inside $F$, so $F=\mathbb{R}[[X]]\oplus D$ as a vector space over $\mathbb{R}$. Choose a vector space isomorphism $C\cong D$. This then induces a vector space isomorphism $S=\mathbb{R}[[X]]\oplus C\cong \mathbb{R}[[X]]\oplus D=F$. Transferring the field structure of $F$ along this isomorphism, we get the desired field structure on $S$.