What is the highest score in a first-class game of cricket, acheived
without any batsman scoring a century?

It appears that the highest score in a first-class cricket match where
no batsman scored a century is 605 by Madhya Pradesh v Haryana at
Rajnandgaon, 13-16 March 1999.

Two recent sources cite this score as the record, though one
incorrectly states the total as 608:

Aslam Siddiqui, "Highest fc totals without an individual 100", Online
posting (2001-07-02 06:11:52 PST)
<rec.sport.cricket> via Google Groups
http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&oe=UTF-8&safe=off&selm=25055b16.0107020511.1ff7aeb%40posting.google.com

"Ask Philip", answer from 15 Jan 2002
CricInfo Global
http://www.cricket.org/link_to_database/INTERACTIVE/ASK_PHILIP/200201/FC_BAT.html

The scorecard is available here:

"Madhya Pradesh v Haryana at Rajnandgaon (MP), 13-16 Mar 1999"
CricInfo: Indian Cricket
http://www.cricket.org/link_to_database/ARCHIVE/1998-99/IND_LOCAL/RANJI/SUPER/GROUP-C/MP_HRYNA_RJI-SL_13-16MAR1999.html

- justaskscott-ga


Search terms used (in various combinations) on Google and Google
Groups:

Many variations on "without a century", such as "no individual
hundred", "without a 100", and "no centuries".
"first class"
fc
cricket
highest
"Madhya Pradesh" Haryana Rajnandgaon

The answer to your question is the July 13 2002 Final match of 
India vs England at Lord's, London. England won the toss and elected
to bat. England made  325 / 5 in 50 overs  (Run Rate 6.5)

India won the match by making India   326 / 8 in 49.3 overs (Run Rate
6.61) the highest scorer in the innings being 87 not out by Mohammed
Kaif. There were no centuries. Here is the link

http://www.thatscricket.com/specials/eng-summer/scorecards/130702match10.html

I hope that answers your question.