Without the temple and access to the ritual of the red heifer, we're all tamei (ritually impure) today with no way to fix it. Yet we work to avoid situations that would make us tamei (e.g. kohanim still don't go near most dead, aside from close relatives). Why is this? Can one become "more" tamei with each exposure? Or does it not have anything to do with our state but, rather, is about not developing bad habits that will be hard to break when we do have a third temple?
Once a Kohain is ritually impure due to contact with dead, is there any prohibition of further impurity? This makes a difference for med students and pulpit rabbis.
The simple understanding is that for non-Kohens, yes we're all tamei so it makes no difference. You want to live in your own special chamber on Jerusalem bedrock, that's fine; you want to be an undertaker, that's fine too.
For Kohens, the prohibition appears to be (and again there may be other opinions, but appears to be): don't come in contact with the dead in any way that could make you tamei. Even if the Kohen just touched corpse #1, he shouldn't touch corpse #2, as each touch is the kind that could make him tamei.
Even for a Kohen, there's no prohibition whatsoever at becoming tamei via lesser sources, such as dead lizards. A Kohen today can touch as many dead lizards as he likes.
(As the story goes, Professor Saul Lieberman once found a thousand-year-old manuscript from someone claiming a Kohen was prohibited to touch dead lizards too; to which Rabbi Joseph Dov Soloveichik responded:
"so what do you think of Professor Jacob Aegis' work?"
"Aegis?! He's a fool! A nincompoop! He has no business in scholarship!"
So you don't think there were people like that a thousand years ago?
An alternative view maintains that really the prohibition is adding levels of tumah, not just performing an act that would make one tamei.
When one touches a dead body he is tamei for at least 7 days. Thus if a kohein would touch a corpse one day, to touch one the next day would be to increase his levels of tumah because he restarts his 7 day count.
Moreover, one who is in contact with a corpse is on a higher level of tumah than one who had touched one. This rule is called "cherev ke-chalal" and is analyzed in the first perek of the Mishnayot in Ahilot. Basically it means while you are touching you become an extension of the corpse such that anything you are touching (for instance, your clothes) become tamei to the first degree, as opposed to second degree, even though they themselves aren't touching the corpse! Thus a kohein who touched a corpse and let go, would be adding to his level of tumah by touching it again.
It is thus very reasonable to argue that a kohein who is currently touching a corpse has no obligation to let go, as he is not raising his tumah level -- it is already risen.
One could discuss whether touching the corpse for a full day is a violation of adding to the 7 days, as above. This is a matter unresolved in the Achronim.
Here is an excellent post on the issue: Kohanim becoming Tamei Nowadays – What’s the Story?
There are many types of unrelated and potentially overlapping Tumah, as well as different degrees of Tumah. Assuming you are talking repeat exposures to the same degree of Tumath Meth:
You can't, and we don't. Kohanim are prohibited from an action that carries a consequence of Tumah, regardless of whether the consequence is in force, which could make it seem otherwise (ie., that you can become more Tamei and we care).