Three fours, five operations, score 1.0413

$$ \sqrt[4!]{4!}+\sqrt{4} \approx 3.141586 $$

_{This doesn’t seem to be the type of question that calls for spoiler hiding, correct me if I’m wrong.}

Three fours, five operations, score 1.0413

$$ \sqrt[4!]{4!}+\sqrt{4} \approx 3.141586 $$

Also five operations but too cute not to include, score 0.7539:

$$ \sqrt[4]{44\sqrt{\sqrt{4!}}} \approx 3.141423 $$

Four operations, score 0.7059:

$$ \sqrt4^\sqrt{\log_4{44}} \approx 3.143093 $$

The best three-operation expression has already been posted in another answer.

For completeness sake, two, score 0.4245:

$$ \left\lceil \log_4{44} \right\rceil = 3.0 $$

and one, score 0.3852:

$$ \log_4{44} \approx 2.729716. $$

_{This doesn’t seem to be the type of question that calls for spoiler hiding, correct me if I’m wrong.}