Interesting topic. So far as we know Eugene Island is the only documented case thus far. There are some theorists who espouse "abiogenic" (without biology) generation of oil but there has not been any confirmation of this as a viable theory to date.
In the case of the Saudis they did a lot of their initial development in Ghwar at the advice of Aramco engineers who told them that ultimate recovery was not Rate Sensitive. In other words, the amount they ultimately recovered would not depend on the rate at which they produced. So they essentially undertook primary, secondary and tertiary recovery simultaneously and were initially hugely pleased since it was relatively cheap to inject volumes of salt (ocean) water in a few locations to keep reservior pressures high and thus avoid the cost of pumping individual wells. Regular vertical wellbores were drilled out horizontally and then "bottlebrushed" out from there. What is happening now is that as these fields begin to water out they do so at a rate that is much faster than in traditionally developed fields. The decline curve isn't a curve, it's a cliff.
So, Ghwar may have peaked, don't know yet for sure, will only know in hindsight. Mexico has peaked, likewise Northsea and not a few others. But into the PEAK discussion has to be factored prices. As prices rise more oil is producible and comes into the market.
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