There are many examples of quantum anomalies that are one-loop exact, and many examples of anomalies that have contributions to all orders in perturbation theory. I haven't been able to identify a pattern though:
Is there any way to tell, from first principles, whether a given (potential) anomaly is $n$-loop exact (for a certain finite $n\in\mathbb N$)? Or does this require a case-by-case analysis?
Is there any example of an anomaly that is $n$-loop exact, for some $n>1$?
Before answering the question, let me remark that number of loops required to obtain the anomaly exactly isn't an invariant quantity; the anomaly is physical, but the number of loops is not.
In the case of chiral anomaly, we need to evaluate a loop diagram because we are using Grassmann fields as coordinates the fermions infinite dimensional configuration space. It is very difficult to do quantum field theory otherwise, but we must remember that the Grassmann fields are just coordinates, thus of no physical significance. The chiral anomaly can be obtained at tree level from the classical Lagrangian of a Weyl particle, i.e., classically, please see the following work by Stone and Dwivedi $^1$. This work is based on the seminal work by: Stephanov and Yin (Chiral kinetic theory).
The essence of their construction can be formulated in quite ingenious simple phase space arguments as given by Kharzeev based on a deep observation by Gribov.
A deep examination of Stephanov and Yin's (or Stone and Dwivedi's) argument shows that the only additional information needed beyond the classical Lagrangian of a Weyl particle is that it obeys the Fermi-Dirac statistics (Here we don't have the Grassmann variables to take care of this part) and that we are working in the infinite volume limit.
Given the above, the exceptional phenomena about the anomaly is that it can be computed exactly (regardless of the number of loops). Certainly, there are topological explanation of this exactness in the case of the chiral anomaly, but the topological arguments do not cover all the cases where quantities can be computed exactly in perturbation theory.
The deep reason is supersymmetry.
In mathematics, this phenomenon is known by equivariant localization, based on the seminal work by Duistermaat and Heckman, please see the following physically oriented review by Szabo. (The rigorous mathematical results exist mainly for the finite dimensional cases; the path integral applications were introduced in the physical literature; they are less rigorous but led to marvelous results especially by Witten).
Basically, localization means that instead of performing the integral over an entire phase space, the result can be obtained by summing the contributions from a smaller subset which can be discrete, or in the case of path integrals can be a finite dimensional manifold (instead of the infinite dimensional path space). Please see the following presentation by Hosomichi. The existence of supersymmetry is responsible for solvability of the $N=2$ supersymmetric gauge theories in $4$ dimensions.
Now, how the chiral anomaly is related to the above: According to Schwinger's proper time method, processes described by excitations of quantum fields can be expressed as quantum mechanical (i.e., in $0+1$ dimensions) path integrals, please see the following review by Bastianelli and van Nieuwehuizen. This is true for example to the process described by the triangle diagram. The quantum mechanical action describes a spinning particle in $0+1$ dimensions which is supersymmetric, therefore can be solved by the localization techniques. This is just what Friedan and Windey did in their seminal work.
$^1$ Although Stone and Dwivedi remark that the alignment of the spin along the angular momentum of a massless particle is a quantum phenomenon; it can be obtained entirely in classical mechanics, please see Duval and Horvathy.
