Non-algebraizable Formal Scheme?

I think the following should work.

Let $X$ be a smooth, complex, projective $K3$ surface, and let $\bar{A}$ be the base of the formal semi-universal deformation of $X$. It is well-known that

$\bar{A}=\mathbb{C}[[X_1, \ldots, X_{20}]]$.

Let $\mathcal{X} \to \textrm{Specf}(\bar{A})$ be the corresponding formal scheme. Then $\mathcal{X}$ is not algebraizable. Roughly speaking, the reason is that the general deformation of $X$ is a $K3$ surface which is $not$ algebraic.

For a complete proof, see [Sernesi, Deformations of algebraic schemes, Example 2.5.12].

EDIT. As it is also remarked in Sernesi's book, this example shows that a smooth, complex, projective variety $X$ need not have an algebraic formally versal deformation, even if the functor $\textrm{Def}_X$ is prorepresentable and unobstructed.

Bhargav's example is really an example of a non-algebraic formal subscheme of the affine plane. Such examples are ubiquitous in foliation theory : a differential equation and, more generally, a foliation on a (smooth) algebraic variety has local leaves which are smooth formal schemes. This follows from the formal Frobenius theorem (in positive characteristic, the foliation needs to have p-curvature zero). Sometimes, these leaves are the formal completions of an algebraic subvariety, but often not. However, these leaves are isomorphic, at formal schemes, to the formal completion at the origin of an affine space; from the intrinsic point of view, they thus are algebraizable.

The theorems of Hironaka, Matsumura, Hartshorne to which Francesco Polizzi refers are in the same spirit, but concern formal subschemes along an algebraic subvariety. They don't apply to formal subschemes based at a point.

Actually, Arakelov geometry allows to establish analogs of these theorems and algebraize some formal subschemes based at a point (eg leaves of a foliation). See papers of Bost (Pub. Math IHES, vol. 93, 2001), and of Bost and myself (Manin Festschrift, 2010).

I find it more or less illusory to ask for non-algebraizable formal schemes which would not fit into the scope of deformation theory. Indeed, a formal scheme $\hat X$ over $C[[t]]$, say, is nothing but a family of schemes $(X_n)$, where $X_n$ is a scheme over $C[t]/(t^{n+1})$ together with isomorphisms of $X_n$ with $X_{n+1}\otimes C[t]/(t^{n+1})$.

On the other hand, I wonder whether classical examples of non-algebraic analytic spaces, or algebraic spaces, could be constructed in the category of formal schemes, but I have no precise answer to give.