Does Largest Target affect only whoever uniquely has the most HP?
This card only affects a player when they have the highest HP and aren't tied with someone else.

So, this is one of those weird phrases in games that's always used but doesn't make any sense as to why it gets used still.  What' we're looking for is the relationship between hp of players.

At the start of the turn, let's say I have 5 and Jim has 6 HP.  Jim will take 1 damage, since he has more hp than the other runner in the game.  At the start of the next round, we both have 5 HP, and nobody takes damage because nobody has more HP than anyone else.

For a more complex example, lets' say I have 4, Tina has 3, Jim has 4, and Al has 2.  In this case, I have more HP than Tin and Al, but not Jim.  I won't take damage.  Jim will say the same, since he's got the same HP as me, and more than Tina and Al.