Depends roughly on the current ability/status of the bowler.
Entry level, recreational, or occasional bowlers don't need more than a single, well-fitting ball. Even bowlers up to rougly a 150 average should stick to one ball and learn mechanics first.
As your skill increases and your average climbs - intermediate level bowlers to an average of say about 180 - may want to carry 2 basic balls. A ball for when the lanes are hooking (less oil) and for when they aren't (more oil). At this level, emphasis should be placed on spare making and a general idea of lane conditions and how to adjust to them. Fiddling with 5,6+ balls isn't going to help much.
Beyond the above, it really depends on what type of bowling you're doing. If you're a one-league-per-week house bowler, or even a twice-per-week league bowler bowling in the same house, the aforementioned two balls are likely more than sufficient. We don't see conditions changing enough in house leagues these days to warrant the need for even 4 balls. A straight spare ball may be the only addition you need.
The next step up, would be bowlers that bowl in two or more different leagues in different houses or are in a travelling league. These bowlers will typically want a ball for wet, a ball for medium, and a ball for dry, and likely (but not necessarily - the dry ball can be used) a straight-going spare ball.
Next are the tournament bowlers who bowl a different house and even sometimes in different states and even different countires, every month, twice a month, or once (and even twice!) per week. These are the bowlers who are going to see a number of different surfaces (old wood, newer wood, older synthetics, newer synthetics), lane conditions, and challenging patterns that often dictate ball selection - and knowledge of drilling and layout and how and why they will or will not work on the various condtions - far beyond "a ball for dry lanes" and "a ball for oily lanes" and "a ball for medium lanes". Even so, these experienced bowlers may only need 1 or 2 balls to be competitive depending on their style of play. However, when you're paying $100 (or more) tournament entry fee twice per month (or even more often) you don't want to get hung out to dry because the lanes are screaming and you didn't buy yourself that $60 weak reactive pearl.
Finally - the collectors, hobbiests (is that even a word?), and enthusiasts with money to burn. For them, the sky is the limit and there is no need to legitimately validate another ball purchase.
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Kill the back row
