I'm thinking something like Texas Holdem or Omaha where one hole card would be displayed at all times.
There are three common methods poker games use to "develop" hands over multiple betting rounds. In order of age of introduction:
The draw. Each player exchanges one or more of his cards for the same number from the stub. Usually these are all hidden cards. Games with only one draw (and therefore two betting rounds) were dominant for many years. Today, games like triple-draw are popular.
Stud. Each betting round, additional private cards are dealt to each player, some face up, some face down. It is common for the initial deal to be two or three cards with only the last one face up. Most subsequent rounds are single face up cards. Often the final one is face down. Many variations here. Some stud games contain a "twist" round, which is actually a draw, in which a player may replace one of his cards with one from the stub, face down for face down or face up for face up.
Community cards. Instead of more cards being dealt to each player, cards are dealt to a "board" in the center of the table, which all players share. Texas Hold'em is the most popular game of this type, and is currently (2014) the most popular type of poker in the US and most of the world (though there are still places like Mexico where stud is the most popular game).
All of these can be mixed to create interesting games. You could, for example, have an initial deal to each player of two down cards and one up, then have a community board, then have a twist.
My personal general rules for making a good game are this: six to eight cards at the end is reasonble. Seven is ideal. Two or three hidden cards is good. Two to five betting rounds is reasonable--I prefer three or four. Limits should be larger for later rounds at limit play.
As the other Jon Suggested hole cards by definition are hidden. The term "Hole Card" in poker means the cards in your hand that are only known to you. So your question may be worded badly.
Having said that there are many forms of poker where one or more of your cards are displayed from the beginning of your hand. None of which are Hold'em or Omaha per say. There are hundreds of variations of playing poker, using a five card hand.
Seven card stud is the most common game using hole cards and up cards. It is found on line in several variations, like hi-lo, and Razz. It is also found in a few casinos.
You can make up your own games if your playing at home, using any combinations of up cards, hole cards, draws and community cards you would like. Your only limitations here are your imagination and the number of players you need to accommodate.
Do any of these games have names, yes they do, names like Guts, which is more a class of poker games that involve large numbers of cards in a players hand, Baseball and Cincinnati. There are books written describing the rules and common names for variations of poker.
Online there are a couple of offerings of unconventional games, It has been awhile (since before the crash) since I have played them. There are also some game sites that offer guts and Cincinnati type games with play money.
I was hoping that when online poker came along that they would start offering many variations of poker online and make poker interesting and challenging again. Sorry to say I was wrong and the simplest variation of poker became dominant.
There is one variation of poker which could be said to meet your requirements: Blind Man's Bluff. In this, you can see everyone's hole card(s). The catch, of course, is that you cannot see your own.
According to the link provided, "During its coverage of the 2004 World Series of Poker, ESPN showed a Blind Man's Bluff version of Texas hold'em" so it has been played in a prime time tournament setting before. I did a quick search on YouTube and there is actually some footage.