Yes. $1 a day is $365 a year — an emergency-buffer starter. More importantly, the habit itself is what compounds: once daily saving is automatic, raising it to $2 or $5 costs nothing psychologically.
$1/day = $365/year. $3/day = $1,095/year. Add a modest 4% annual return over 10 years and $3/day becomes roughly $13,400. The number is real; the behavior is the hard part.
Small commitments avoid the "all-or-nothing" trap. Behavioral research (Thaler, Benartzi) shows people stick to plans they barely notice, and quit plans that hurt.