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.