  We define a model for random (abstract) cell complexes (CCs), similiar to the
well-known Erd\H{o}s-R\'enyi model for graphs and its extensions for simplicial
complexes. To build a random cell complex, we first draw from an
Erd\H{o}s-R\'enyi graph, and consecutively augment the graph with cells for
each dimension with a specified probability. As the number of possible cells
increases combinatorially -- e.g., 2-cells can be represented as cycles, or
permutations -- we derive an approximate sampling algorithm for this model
limited to two-dimensional abstract cell complexes. Since there is a large
variance in the number of simple cycles on graphs drawn from the same
configuration of ER, we also provide an efficient method to approximate that
number, which is of independent interest. Moreover, it enables us to specify
the expected number of 2-cells of each boundary length we want to sample. We
provide some initial analysis into the properties of random CCs drawn from this
model. We further showcase practical applications for our random CCs as null
models, and in the context of (random) liftings of graphs to cell complexes.
Both the sampling and cycle count estimation algorithms are available in the
package `py-raccoon` on the Python Packaging Index.
