There's a CD in print right now called "Best of Arthur Godfrey: The Old Redhead" Collectors' Choice Music #177. Here are the contents, according to the All Music Guide:

1. Too Fat Polka (I Don't Want Her, You Can... (MacLeah/Richardson) - 3:01
2. A Porter's Love Song to a Chambermaid (Johnson/Razat) - 2:51
3. Slap 'Er Down Again, Pau (Arnold/Asherman/Cornett) - 2:57
4. The Thousand Islands Love Song (Oh... (Hillard/Sigman) - 2:44
5. California Is Wonderful (If You're a... (David/Hoffman/Livingston) - 2:52
6. I'm Looking Over a Four-Leaf Clover (Dixon/Woods) - 2:47
7. All Right, Louie, Drop the Gun (Carter/Johnson) - 3:08
8. Go to Sleep, Go to Sleep, Go to Sleep (Cahn/Spielman) - 3:21
9. Gone Fishin' (Kenny/Kenny) - 2:31
10. C'est Tout (That's All) (Dorsey/Elmer/Geva-Ayers) - 3:00
11. The Thing (Grean) - 2:26
12. I Like the Wide Open Spaces (Foster/Murray/Wick) - 2:38
13. Love Is the Reason (Fields/Schwartz) - 2:52
14. Slow Poke (King/Price/Stewart) - 2:36
15. Dance Me Loose (Erwin/Howard) - 2:48
16. Exactly Like You (Fields/McHugh) - 1:55
17. (We Grow) Too Soon Old-Und Too Late Shmart (Drake/Shirt) - 2:25
18. The Humour Is on Me Now (Hayward) - 2:01
19. Sidewalks of New York (Balke/Lawlor) - 2:31
20. Seventy-Six Trombones (Wilson) - 2:36
21. Pale Potomac Moon (Godfrey) - 2:26
22. Young at Heart (Leigh/Richards) - 2:15

A few of these are fairly familiar and several more have intriguing titles.

By the way, as much as I admire AMG for being a source of wonderful information like this, I must admit that some things I see there are totally bogus. F'rinstance, it classifies the above album as "Rock." Then under the heading "Similar/Related Albums" it lists Crosby, Stills, Nash, & Young, and Tracy Chapman!