as of now I don't own a single copper coin. I wouldn't pay a big premium for full RD copper coins b/c I think they'll turn into RB coins in the future.
I don't own any classic head cent or half cents. They've been pretty high on my wish list and now that I've completed my braided hair set (minus a few that will likely get upgraded) I think I'll go after the half cents next. Either that or three cent silvers. I have a couple of proofs, but I've never gone after any business strikes.
There are too many US coins to mention that I don't own but I'll name two. I'd like to own a 1916 Standing Liberty Quarter, and since I'm dreaming here, make it a high grade one too. Also, I'd like to own a ten dollar gold coin, any one of them but I'd prefer the an Indian Head.
Dancing Fire I'd rather have a brown copper over red any day. I'd rather have any toned coin over "bright and shiny" anyday. Even a coin with "hideous corrosion" is as a wild animal that lived free rather than captured and domesticated in some circus.
Have never owned a shield nickel, just never found one that I like. It has to have a sharp, fully struck look, BU and without planchet flaws. Apparently that is asking too much.
Since my prime collection is a US Type Set I have (almost) one of every Cu/Ni/Ag business strike coin design through 1964. As for Au coins I have one of every business strike design from 1834 through 1933.
Coin Collectors are a different breed, no doubt. Myself, I collect everything. Proof sets, Mint sets, older and new gold coins, errors, Casino chips, casino dollars, Commemorative Halfs, tokens, Civil War coins, Canadian, other Foreign coins and so on. I have just about everything American from the 1900's except the Liberty standing Quarter collection and a few Commemorative Halfs . I am missing 11 Liberty Standing Quarters, mostly because I am completing a very nice set. I have a very nice 1916 and 23-S. However, my dream coin is the 1895 Proof Morgan, that would complete a set of awesome Morgan Dollars.
BUT to answer the original question there are no "popular US coins" that either: -- I don't own, or -- that I want. Of course each person has a different definition of "popular".
This one's tough to answer because it's tough to define popular. Is the $4 gold Stella a popular coin? How about a Chain cent? How about Seated Dollars? These coins garner considerable prices in AU/MS condition. They are classic coins, but are they popular? Tough to say. There are others, but I thought I'd keep the list short. One day, I hope to own all of them as I'm building a Type set.