For half, I could use semi, demi, or hemi. While semi does mean "half", it sometimes has a connotation of "some". Demi is often found with French roots. According to this link, hemi is the least often used of the three but can also mean "half".
On the other hand, I could use something like toti or perhaps other choices to indicate all.
Pardon my mathematical nature, but is there a prefix which would mean "mostly", that is more than one half but less than the full unit.
Depending on context quasi could work. Quasi as a prefix indicates similarity or resemblance. If a sphere is a ball and a hemisphere is half a ball, quasi-sphere would mean something that's ball-like, "roughly" or "more or less" a ball.
While semi does mean "half", it sometimes has a connotation of "some".
I'm not convinced by this. It's sometimes used informally to mean "somewhat" or "sort of", but to mean "some" would be rare.
We have quasi- for "almost", but not for "most of". It's appropriate when something bears a resemblance to another thing, rather than when it is an incomplete example of something.
We don't have toti- as a common prefix, but we do have pan- for all of, which is used productively.
pene-, a Latin root, meaning almost, such as in the word penultimate
Although I'm rather sure pene- is prefix you want, here are a few other prefixes that are close in meaning, including some Greek if you were preferring a Greek root.
mero-, a Greek root, meaning partial
plesio-, a Greek root, meaning near
sub-, a Latin root, meaning under
semi-, a Latin root, meaning, partial, half
I'd use omni- or multi-. While omni- indeed means all, it often is appropriated for use as mostly, as for example so-called omnidirectional microphones and antennae are not indeed omnidirectional, but instead most-directions–directional. Multi-, on the other hand, is less of a stretch, as it denotes “Resembling, pertaining to many or many things”.
