After seeing wood's last comment (comment #2 under his question), I've decided to add a few words (a bit too many for a comment) which hopefully make clear the force of Qiaochu's answer.

Generally speaking, the categorical meaning of "completion" refers to taking a left adjoint of a *full* inclusion of categories; in our situation we are considering the full subcategory

$$\text{Conn} \hookrightarrow \text{Top}$$

from connected spaces to general topological spaces. Examples where such completions exist are: the inclusion of complete metric spaces into the category of metric spaces and continuous maps with Lipschitz constant 1 (Cauchy completion), the inclusion of compact Hausdorff spaces into the category of all spaces (Stone-Cech compactification), and the inclusion of fields into the category of integral domains and injective ring maps (field of fractions construction).

(There's a bit of fine print here: sometimes one also demands that the unit of the adjunction, here the universal map of an object to its completion, be injective. For example, the inclusion of abelian groups into the category of groups does have a left adjoint (the abelianization), but this isn't injective. Similarly, to get the map from a space to its Stone-Cech compactification to be injective, one should really consider the inclusion of compact Hausdorff spaces in the category of completely regular spaces. Sometimes the suffix -ization or -ification is used in cases where the unit is not injective.)

The salient point behind Qiaochu's answer is that a left adjoint, if it exists, must preserve coproducts (or in fact colimits generally). Now, supposing that the left adjoint to the inclusion $\text{Conn} \to \text{Top}$ exists, it would first of all take a one-point space to a one-point space (the proof is easy), and it would take a coproduct of two one-point spaces in $\text{Top}$, viz. a two-point discrete space, to a coproduct of two one-point spaces in $\text{Conn}$. But Qiaochu's example shows this cannot possibly exist.

The only remedy that I can think of in this situation is to change things up a bit, in a way that I don't think will be at all useful to the OP. There are for example situations where an algebraic structure on a space forces it to be connected (and then some), where one can construct the corresponding free algebraic structures to get a left adjoint to the (non-full) forgetful functor mapping to $\text{Top}$. The most obvious example might be to consider spaces equipped with a contraction: consider spaces $X$ equipped with a basepoint $x_0: 1 \to X$ and with an action $\alpha: [0, 1] \times X \to X$ of the multiplicative monoid $[0, 1]$, such that $\alpha(0, x) = x_0$ for all $x \in X$. Here the free algebra on a general space $Y$ is just the cone $CY$ with the obvious algebraic structure. But this isn't likely to be useful to the OP.