The total number of tunes played at Woodstock according to my listing below is 298
However according to "The Woodstock Project" there were around 287 songs performed, I am not for one second saying they are wrong.
It may depend on your interpretation of an event. Is a medley of 3 songs, 3 songs or just 1 ?
Sly and the Family Stone's set had one medley that lasted 22 minutes. It may also depend on what you concider a song. Was the intro music for Sha-Na-Na a song or just a ditty that shouldn't be included. The Incredible String Band gave us a rendition of a poem entitled "Invocation", should this be included or left out. The same could be said of the Guru Swami Satchadinanda, he did perform at Woodstock (he came on just after Richie Havens on Friday) but is invariably left out of any set list.
It is all a little hazy around the edges, much like the festival that took place 54 years ago.
Like the old saying goes: If you remember the 60's, you weren't really there :)

Cancelled Acts

1. Jeff Beck Group (The band broke up in July, forcing their cancellation)
2. Iron Butterfly (Were stuck at an airport, their manager demanded helicopters and special arrangements just for the group.
   The manager was wired back and told, as impolitely as Western Union would allow, "to get lost".
3. It's a Beautiful Day (dismissed)
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Acts without music

1. "The Joshua Light Show" were at Woodstock but were a light show rather than a band so no track listing
2. "Swami Satchidananda" was at Woodstock but was an Indian guru, so no track listing
3. "The Big Storm" was at Woodstock but was a weather condition not a band, so no track listing
4. "Max Yasgur" had to be at Woodstock, it was his farmland, no track listing
5. "Abbie Hoffmann" was on stage if only for a short time ;)
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The Woodstock Music and Art Fair
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Although officially listed as a three day event and tickets sold appropriately, it was in fact held over four days, from 5.07pm on Friday August 15th 1969 until 10.30am on Monday August 18th 1969.
The above set listings is as accurate as I can possibly get it with each artist assigned to the logical (or as logical as makes sense) dating. I have arranged each artist into the day they actually started playing. So, any artist that began playing at midnight or after has been assigned to the following days set listing, with the exception of Arlo Guthrie and Joan Baez, who started performing after midnight (Saturday morning) but were the last acts for 8 hours.

You will notice that in the official set listings (below) Joe Cocker began day three. However if you look at my set listing (above), Jefferson Airplane and a few other artists played the same day but in the morning rather than the afternoon, where's the logic?

Day 1 Friday August 15th)
Richie Havens
Sweetwater
Bert Sommer
Tim Hardin
Ravi Shankar
Melanie
Arlo Guthrie
Joan Baez

Day 2 (Saturday August 16th)
Quill
Country Joe
Santana
John Sebastian
Keef Hartley
Incredible String Band
Canned Heat
Mountain
Creedance Clearwater Revival
The Grateful Dead
Janis Joplin
Sly & the Family Stone
The Who
The Jefferson Airplane

Day 3 (Sunday August 17th)
Joe Cocker
Country Joe & the Fish
Ten Years After
The Band
Johnny Winter
Blood Sweat & Tears
Crosby, Stills, Nash, and Young
Paul Butterfield Blues Band
Sha-Na-Na
Jimmy Hendrix

Approximately (no official records have ever come to light) 298 tunes (287 according to the Woodstock Project) were played throughout the event.
Some artists namely: Ravi Shankar, Santana, Mountain, Grateful Dead, Creedance Clearwater Revival, Janis Joplin, Sly & the Family Stone, The Who, Jefferson Airplane, Johnny Winter, Crosby, Stills, Nash & Young, and Jimi Hendrix are known to be complete sets. The other artists sets are more than likely complete but not guaranteed as such.