I was given this problem 30 years ago by a coworker, posted it 15 years ago to rec.puzzles, and got a solution from Barry Wolk, but have never seen it again. Consider the series:
$1, \frac{1}{2},\frac{\displaystyle\frac{1}{2}}{\displaystyle\frac{3}{4}},\frac{\displaystyle\frac{\displaystyle\frac{1}{2}}{\displaystyle\frac{3}{4}}}{\displaystyle\frac{\displaystyle\frac{5}{6}}{\displaystyle\frac{7}{8}}},\cdots$
Each fraction keeps its large bars while being put atop a similar structure.
This can also be represented as $\frac{1\cdot 4 \cdot 6 \cdot 7 \cdot\cdots}{2 \cdot 3 \cdot 5 \cdot 8 \cdot\cdots}$ terminating at $2^n$ for some $n$, where it is much closer to the limit than elsewhere.
The challenge:
Find the limit, not too hard by experiment
In the last expression, find a simple, nonrecursive, expression to say whether $n$ is in the numerator or denominator
Prove the limit is correct-this is the hard one.