Must the set of lines through the origin on which a nonconstant entire function is bounded be finite?
Newman gave an example in 1976 of a non-constant entire function bounded on each line through the origin in "An entire function bounded in every direction".
I like the second sentence of the article:
This is exactly what is needed to confuse students who have just struggled to comprehend the meaning of Liouville's theorem.
Armitage gave examples in 2007 of non-constant entire functions that go to zero in every direction in "Entire functions that tend to zero on every line". For this I have only seen the MR review. (If you don't have MathSciNet access, the link should still give you the publication information to find the article.)
Update: I just decided to take a look at the Armitage paper, and the introduction was enlightening:
Although every bounded entire (holomorphic) function on $\mathbb{C}$ is constant (Liouville’s theorem), it has been known for more than a hundred years that there exist nonconstant entire functions $f$ such that $f(z) → 0$ as $z →∞$ along every line through 0 (see, for example, Lindelöf’s book [10, pp. 119–122] of 1905). And it has been known for more than eighty years that such functions can tend to 0 along any line whatsoever (see Mittag-Leffler [11], Grandjot [8], and Bohr [4]). Further references to related work are given in Burckel’s review [5] of Newman’s note [12]. Entire functions with radial decay are used by Beardon and Minda [3] and Ullrich [14] in studies of pointwise convergent sequences of entire functions.
Armitage goes on to mention that Mittag-Leffler and Grandjot also gave explicit constructions, but states, "The examples given in what follows may nevertheless be of some interest because of their comparative simplicity." The examples are $$F(z)=\exp\left(-\int_0^\infty t^{-t}\cosh(tz^2)dt\right) - \exp\left(-\int_0^\infty t^{-t}\cosh(2tz^2)dt\right)$$ and $$G(z)=\int_0^\infty e^{i\pi t}t^{-t}\cosh(t\sqrt{z})dt\int_0^\infty e^{i\pi t}t^{-t}\cos(t\sqrt{z})dt .$$
The Mittag-Leffler function $E_{\alpha,1}$, $\alpha>0$, is bounded in the sector $$\frac{\alpha\pi}{2}< \arg z<2\pi-\frac{\alpha\pi}{2}.$$
In particular, $e^z=E_{1,1}(z)$ is bounded in $$\frac{\pi}{2}< \arg z<\frac{3\pi}{2}.$$