  We study Fibonacci compositions, which are compositions of natural numbers
that only use Fibonacci numbers, in two different contexts. We first prove
inequalities comparing the number of Fibonacci compositions to regular
compositions where summands have a maximum possible value. Then, we consider
asymptotic properties of Fibonacci compositions, comparing them to compositions
whose terms come from positive linear recurrence sequences. Finally, we
consider analogues of these results where we do not allow the use of a certain
number of consecutive Fibonacci numbers starting from $F_2 = 1$.
