What is the maximum possible number of lives?
Solution 1:
These are the items that grants extra lives (only those with a graphical feedback on the lives counter):
- Dead cat (9 lives)
- 1up! (1 life)
- Ankh (1 life as ???)
- Lazarus' Rags (1 life as Lazarus)
- Judas' Shadow (1 life as Black Judas)
- Guppy's Collar (50% chance of 1 life)
The total will make 13?.
There's a way to increase that total, using Diplopia 
(Afterbirth exclusive) on Dead Cat. In this way you can get another Dead Cat, therefore obtaining 18 extra lives.
(Assuming you can use Diplopia only once in a run.)
Adding the remaining items will make an amazing total of 22? lives!
EDIT:
As EnragedTanker suggested, if you manage to have every extra-life item in one room (using some kind of game-breaking exploit like Blank Card + Jera) and then use Diplopia on them you will double each item.
In this case the total will be 26? lives.
Solution 2:
That depends on what you mean.
Programming-wise, you can get an infinite number of lives if, for example, you just picked up a whole bunch of Dead Cats. In a normal game this isn't possible, but with cheats/mods you could.
Theoretically, Guppy's Collar can bring you back an infinite number of times as well. The probability of getting more than two or three lives out of it is pretty low, but still, it's possible to get hundreds, thousands, even millions of lives out of that item. Same thing with the Broken Ankh.
However, if we're talking practically, then the maximum number of lives that can appear on the life list is directly related to the number of life-reviving items in the game. In Rebirth, the maximum would be "x14?", which can happen while playing as Lazarus and collecting the following items: 1-Up, Ankh, Dead Cat (x9), Guppy's Collar, Judas' Shadow, and Lazarus' Rags. Additionally, while carrying the Broken Ankh trinket, you also have an additional 22% chance of reviving as Blue Baby once all your other lives run out.
In Afterbirth, you can also collect the Mysterious Poster trinket, which will revive you as the Lost when you die. This effectively bumps the max to "x15?", though the on-screen number will not change.