We show that large rectangular semigroups can be found in certain Stone-Čech compactifications. In particular, there are copies of the
2
c
×
2
c
2^{\mathfrak {c}}\times 2^{\mathfrak {c}}
rectangular semigroup in the smallest ideal of
(
β
N
,
+
)
(\beta \mathbb {N},+)
, and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of
(
β
N
,
+
)
(\beta \mathbb {N},+)
if and only if it is a subsemigroup of the
2
c
×
2
c
2^{\mathfrak {c}}\times 2^{\mathfrak {c}}
rectangular semigroup. In fact, we show that for any ordinal
λ
\lambda
with cardinality at most
c
\mathfrak {c}
,
β
N
\beta {\mathbb {N}}
contains a semigroup of idempotents whose rectangular components are all copies of the
2
c
×
2
c
2^{\mathfrak {c}}\times 2^{\mathfrak {c}}
rectangular semigroup and form a decreasing chain indexed by
λ
+
1
\lambda +1
, with the minimum component contained in the smallest ideal of
β
N
\beta \mathbb {N}
. As a fortuitous corollary we obtain the fact that there are
≤
L
\leq _{L}
-chains of idempotents of length
c
\mathfrak {c}
in
β
N
\beta \mathbb {N}
. We show also that there are copies of the direct product of the
2
c
×
2
c
2^{\mathfrak {c}}\times 2^{\mathfrak {c}}
rectangular semigroup with the free group on
2
c
2^{\mathfrak {c}}
generators contained in the smallest ideal of
β
N
\beta \mathbb {N}
.