I haven't done a detailed count, but I think Irving Berlin has a strong claim to be the composer/song writer with most songs recorded by Bing.

An interesting centenary occurs this year. In 1914 Berlin's first show "Watch Your Step" opened in New York. The show was for Vernon and Irene Castle, the dance team then taking things by storm, and later the subject of a movie with Fred Astaire and Ginger Rogers "The Story Of Vernon And Irene Castle". One of the songs featured was "Play A Simple Melody".

Berlin had already made a name for himself with "Alexander's Ragtime Band", in 1911.

Bing recorded (or performed on film, radio and TV) numerous Berlin songs, and this might not be an exhaustive list - 

Abraham - Alexander's Ragtime Band -All Alone - All By Myself - All Of My Life -Always - Angels Of Mercy - Anything You Can Do -

Be Careful, It's My Heart - The Best Thing For You - Blue Skies -

Call Me Up Some Rainy Afternoon - The Call Of The South -Change Partners - Cheek To Cheek - Coquette  - Count Your Blessings Instead Of Sheep - A Couple Of Song And Dance Men -

Easter Parade -Everybody Step - 

A Fella With An Umbrella - The Freedom Train -

Gee, I Wish I Was Back In The Army -(Running Around In Circles) Getting Nowhere - God Bless America -

Happy Holiday - Heat Wave - How Deep Is The Ocean? -

I Got The Sun In The Morning -I'd Rather See A Minstrel Show -I'll Capture Her Heart - I'll See You IN C.U.B.A. - I'm Playing With Fire - I'm Putting All My Eggs In One Basket - Isn't This A Lovely Day To Be Caught In The Rain? - I've Got My Captain Working For Me Now - I've Got My Love To Keep Me Warm - I've Got Plenty To Be Thankful For -

Kate -

Lazy - Let Yourself Go -Let's Face The Music And Dance - Let's Start The New Year Right - The Little Things In Life - 
 A Man Chases A Girl (Until She Catches Him) -Mandy - Marie - Marrying For Love -

Not For All The Rice In China - Now It Can Be Told -  

Oh How I Hate To Get Up In The Morning - The Old Man -

Play A Simple Melody - A Pretty Girl Is Like A Melody -Putting On The Ritz -

Remember - Russian Lullaby -

Say It Isn't So - Say It With Music - A Serenade To An Old Fashioned Girl - Slumming On Park Avenue - Snow - Soft Lights And Sweet Music -Some Sunny Day - The Song Is Ended - Song Of Freedom - Sunshine -

There's No Business Like Show Business - They Say It's Wonderful - This Is A Great Country - This Is The Army, Mr Jones - Top Hat, White Tie, And Tails -

Waiting At The End Of The Road - What Can You Do With A General? - What'll I Do? - When I Leave The World Behind - When I Lost You - When The Folks High-Up Do The Mean Low Down - When The Midnight Choo Choo Leaves For Alabam -

You Keep Coming Back Like A Song - 
And, mustn't forget, White Christmas