2 to the power of 581369, 3 to the power of 366803, and 5 to the power of 250382, are very close.

Not impressed? How about:

8502 tropical years, 105155 synodic months, and 3105289 solar days, are very close.

The bottom line: approximating one real number by rational numbers
is easy, Euclid's (continued fraction) algorithm takes care of that.
But simultaneously approximating several real numbers by rationals of
the same denominator, or similar problems, is a *much* more
difficult problem. So while it is easy to find a power of 2 and a
power of 3 which are very close (try 2 to the 1054 and 3 to the 665,
for example), or a power of 3 and a power of 5, or a power of 2 and a
power of 5, on the other hand, finding all three powers which are
close is very difficult.

Just dabbling, really. Much study has certainly been done on the question, and I know none of it; I just experimented with a few obvious techniques. I guess I should rather stick to my (real) research work.