After one half life, what fraction of the parent element has decayed to a daughter product?
The answer is written in the question - one half!
The term half-time is defined to be the amount of time required for one half of a radioactive material to undergo whatever type of decay is involved, meaning that one-half of the original sample remains!
After a second half-life, you have one-half of one-half (i.e. one quarter) remaining, and so on.
Half life can be extremely long, as in the case of the alpha decay of uranium 238, or extremely short (something on the order of 3 s in the case of Lawrencium, if memory serves me right).