  An Engel series is a sum of reciprocals of a non-decreasing sequence $(x_n)$
of positive integers, which is such that each term is divisible by the previous
one, and a Pierce series is an alternating sum of the reciprocals of a sequence
with the same property. Given an arbitrary rational number, we show that there
is a family of Engel series which when added to it produces a transcendental
number $\alpha$ whose continued fraction expansion is determined explicitly by
the corresponding sequence $(x_n)$, where the latter is generated by a certain
nonlinear recurrence of second order. We also present an analogous result for a
rational number with a Pierce series added to or subtracted from it. In both
situations (a rational number combined with either an Engel or a Pierce
series), the irrationality exponent is bounded below by $(3+\sqrt{5})/2$, and
we further identify infinite families of transcendental numbers $\alpha$ whose
irrationality exponent can be computed precisely. In addition, we construct the
continued fraction expansion for an arbitrary rational number added to an Engel
series with the stronger property that $x_j^2$ divides $x_{j+1}$ for all $j$.
