The following is from the comments section of the original post. I thought it was too good to get lost in the comments:

Any pitchers with a career ERA of infinity (i.e. pitched once or so, gave up a run and no outs) ?

—Robert

It was done 13 times by the following men:

Lou Bauer
Fred Bruckbauer
Bill Childers
Vic Davalillo
Frank Dupee
Bill Ford
Doc Hamann
Will Koenigsmark
Mike Palagyi
Dave Pierson
Jim Schelle
Billy Sunday
Gordie Sundin

Davalillo is the last person to do it and he's the only one to pitch in two games. He faced four batters, two of which walked and two got hits. The most runs allowed without registering an out was 6 by Bill Childers (2 H, 6 BB, 3 WP) in 1895 and Doc Hamann (3 H, 3 BB, 1 HBP, 1 WP) in 1922.

There are also 15 pitchers with unresolved ERAs, meaning that they allowed no runs and did not retire a batter. Among them is Stan Musial, who faced one batter in 1952. Musial started his minor-league career as a pitcher but was forced to switch to a position player after an injury. Here's the account of the game from Baseball-Library.com:

September 29, 1952: Stan Musial makes his only ML pitching appearance. With his 6th batting title wrapped up, he takes the mound against the Cubs Frank Baumholtz, the runner-up in the batting race. Baumholtz, batting righthanded, reaches base on an error, and Harvey Haddix relieves Musial. The Cubs win 3-0 behind Paul Minner.

By the way, Musial and Larry Yount (Robin's brother) are the only two men to appear in a game as a pitcher without registering an out or giving up a hit, walk, or hit batsman. Yount actually never faced a batter. From Baseball Library:

September 15, 1971: At Houston, the Braves win, 4–2, as Aaron belts his 44th homer of the year. The RBI is his 1,954th, tying him with Ty Cobb for 3rd place on the all-time list. In the 9th inning, Larry Yount is announced as the new pitcher for Houston. While taking his warmup pitches, Yount injures his arm and must be replaced in what proves to be his only ML appearance. Larry's brother Robin Yount will begin a more successful ML career in 1974.

Joe "Fire" Cleary holds the highest career ERA for a pitcher who registered an out: 189.00 from 1/3 inning of work with the 1945 Senators in which he allowed 7 earned runs on five hits, 3 BBs, and one wild pitch. The league park-adjusted average ERA was 2.70 that year. His adjusted ERA is a 1.