Borel (1972, J. Diffl. Geometry) proved that $f$ is always algebraic if $Y$ is the quotient of a bounded symmetric domain by a torsion-free arithmetic subgroup. This is a super-generalization of your example 3 (the quotient of the complex upper half plane by $\Gamma(2)$ is isomorphic to the projective line minus three points). The proof uses a generalization of work of Kwack plus the resolution of singularities. 

Added: Kwack (1969) generalized the big Picard theorem by proving that any
holomorphic map from the punctured unit disk into a hyperbolic complex space
can be extended holomorphically to the whole unit disk. [A reduced complex
space is said to be hyperbolic if the Kobayashi pseudodistance is a distance
(Kobayashi 1967).]

Borel 1972 replaced the punctured disk in Kwack's theorem with a product of
punctured disks and disks. Resolution of singularities allows you to realize a
smooth algebraic variety as an open subvariety of a smooth projective variety
in such a way that the boundary is a divisor with normal crossings (hence
analytically a product of punctured disks and disks).

References:

Borel, Armand. Some metric properties of arithmetic quotients of symmetric
spaces and an extension theorem.  J. Differential Geometry 6 (1972), 543--560.

Kwack, Myung H., Generalization of the big Picard theorem. Ann. of Math. (2)
90 1969 9--22. 

Kobayashi, Shoshichi, Invariant distances on complex manifolds and holomorphic
mappings. J. Math. Soc. Japan 19 1967 460--480.