an entire function
An entire function is a function that is holomorphic on the ENTIRE complex plane — there is no point anywhere where it fails to be complex-differentiable, no singularity to avoid. It is the cleanest possible kind of holomorphic function, defined and well-behaved everywhere at once.
The standard examples are friendly: every polynomial in z is entire, and so are e^z, sin z, cos z, and sums, products, and compositions of entire functions. What you must NOT include is anything with a denominator that can vanish (1/z fails at 0), a logarithm or fractional power (these need branch cuts), or a conjugation (z-bar is nowhere differentiable). An entire function has a single power series sum a_n z^n that converges for ALL z (infinite radius of convergence).
Being entire is a strong straitjacket. Liouville's theorem says a BOUNDED entire function must be constant — so any nonconstant entire function (like e^z or sin z) must blow up somewhere as |z| grows, which is why, for instance, sin z is unbounded in the complex plane even though it is bounded on the real line. The growth rate of entire functions (their order and type) is a whole rich theory in its own right.
e^z is entire: its series 1 + z + z^2/2! + z^3/3! + ... converges for every z. By contrast 1/(z - 1) is holomorphic on C minus {1} but NOT entire, because of its singularity at z = 1.
Entire = holomorphic everywhere = power series with infinite radius of convergence.
A nonconstant entire function cannot be bounded (Liouville), so 'entire' never means 'tame at infinity' — sin z and cos z are entire yet unbounded in the plane.