  We consider a point-to-point communication system, where in addition to the
encoder and the decoder, there is a helper that observes non-causally the
realization of the noise vector and provides a (lossy) rate-$R_{\mbox{\tiny
h}}$ description of it to the encoder ($R_{\mbox{\tiny h}} < \infty$). While
Lapidoth and Marti (2020) derived coding theorems, associated with achievable
channel-coding rates (of the main encoder) for this model, here our focus is on
error exponents. We consider both continuous-alphabet, additive white Gaussian
channels and finite-alphabet, modulo-additive channels, and for each one of
them, we study the cases of both fixed-rate and variable-rate noise
descriptions by the helper. Our main finding is that, as long as the
channel-coding rate, $R$, is below the helper-rate, $R_{\mbox{\tiny h}}$, the
achievable error exponent is unlimited (i.e., it can be made arbitrarily
large), and in some of the cases, it is even strictly infinite (i.e., the error
probability can be made strictly zero). However, in the range of coding rates
$(R_{\mbox{\tiny h}},R_{\mbox{\tiny h}}+C_0)$, $C_0$ being the ordinary channel
capacity (without help), the best achievable error exponent is finite and
strictly positive, although there is a certain gap between our upper bound
(converse bound) and lower bound (achievability) on the highest achievable
error exponent. This means that the model of encoder-assisted communication is
essentially equivalent to a model, where in addition to the noisy channel
between the encoder and decoder, there is also a parallel noiseless bit-pipe of
capacity $R_{\mbox{\tiny h}}$. We also extend the scope to the Gaussian
multiple access channel (MAC) and characterize the rate sub-region, where the
achievable error exponent is unlimited or even infinite.
