Program Sequence Generator
JavaScript (ES6), score 77 74 66
The period of this PSG is 5.  Each iteration of the sequence removes one asterisk in the block comment until it reaches 2, then goes back to 6.
f=_=&gt;/**/`f=${f}`.replace('**/','*/').replace('/*\/','/******/')

Let's look at some partial sums to see what the limit approaches.
m     1     2     3     4     5
score 64    66    66.33 66.25 66

Edit: thanks to @ETHproductions for pointing out that I should put the comment first.  Not only is the code much shorter, it appears neater!