  We consider a conservative ergodic measure-preserving transformation $T$ of
the measure space $(X,\mathcal{B},\mu)$ with $\mu$ a $\sigma$-finite measure
and $\mu(X)=\infty$. Given an observable $g:X\to \mathbb{R}$, it is well known
from results by Aaronson that in general the asymptotic behaviour of the
Birkhoff sums $S_Ng(x):= \sum_{j=1}^N\, (g\circ T^{j-1})(x)$ strongly depends
on the point $x\in X$, and that there exists no sequence $(d_N)$ for which
$S_Ng(x)/d_N \to 1$ for $\mu$-almost every $x\in X$. In this paper we consider
the case $g\not\in L^1(X,\mu)$ assuming that there exists $E\in\mathcal{B}$
with $\mu(E)<\infty$ and $\int_E g\,\mathrm{d}\mu=\infty$ and continue the
investigation initiated in previous work by the authors. We show that for
transformations $T$ with strong mixing assumptions for the induced map on a
finite measure set, the almost sure asymptotic behaviour of $S_Ng(x)$ for an
unbounded observable $g$ may be obtained using two methods, adding a number of
summands depending on $x$ to $S_Ng$ and trimming. The obtained sums are then
asymptotic to a scalar multiple of $N$. The results are applied to a couple of
non-regular continued fraction algorithms, the backward (or R\'enyi type)
continued fraction and the even-integer continued fraction algorithms, to
obtain the almost sure asymptotic behaviour of the sums of the digits of the
algorithms.
