So indeed we will never have 21 million coins that way.
You can just ignore the nonsense from Charles-Tim, franky1, and hugeblack. Clearly they didn't pay attention to what you were actually asking (although at least franky1 did point out how at a technical level the "halving" actually works and why the extra half a satoshi simply disappears).
As o_e_l_e_o has pointed out, and you have come to realize, it was never intended that there would be exactly 21,000,000 Bitcoins. The more accurate phrase would be that there will never be more than 21,000,000 bitcoins, since no matter what the number is it will always be less than that. There have on occasion been miners (or mining pools) that have failed to create the maximum allowed block reward in their blocks. Therefore, the actual total will end up being less than the 20,999,999.9769 Bitcoins that were theoretically possible when Bitcoin first started.
though technically there are no btc..
here is the real technical
bitcoin at binary base level has no btc. its units of measure for the human translation of 50btc is
100101010000001011111001000000000 units of measure produce every block in 2009-2011
however some silly people in the development team stupidly think that it is ok to make it so that
although in
2009-2012 100101010000001011111001000000000 (50)
2012-2016 10010101000000101111100100000000 (25)
2016-2020 1001010100000010111110010000000 (12.5)
2020-2024 100101010000001011111001000000 (6.25)
these silly people want to make it so that there are 4 more decimals (12 instead of
meaning changing the hard rule to become
6.25btc 100101010000001011111001000000
6.25btc 1011010111100110001000001111010010000000000
which can break bitcoin in many ways about backward compatibility and forward compatibility of reading transaction values and reward rule amounts
but this also means 13 more halvings. not only changing how many decimals of "btc" coins are produced as of final halving.
but also change the end date of final halving by 52 years and also changing the maximum UNITS OF SHARABILITY (aka sarcity)
so while danny and o_e_l_e_o in some subsections of the forum pretend the max cap is just under 21mill coins and pretend to be smart by mentioning the limit in precise decimal amount. in other sections of the forum they promote breaking these rules to be compatible with other networks.. funnily then breaking their own math of how many decimals will be final..