That sound like what mine is doing. Pretty sure the dealer knew it was doing it too. now that I think about it... Will have to look into getting it fixed. and spend a few minutes in the car after I turn it off, to be sure it doesn't start again.

 

this will conflict with some of the answers given but i have recently had a similar issue but the circumstances may shed extra light.
2001 rx300 / no remote start / no push button,fob,- key only .
i was changing the battery, left the keys in the ignition but in the off position, i loosened the positive cable slightly and was loosening the negative cable (planning to hook up jumpers to my truck battery so i don't lose radio and seat programing but didn't have jumpers hooked up at this point) loosened neg cable slightly and car started by itself - i was outside so obviously i can't have my foot on the brake. i got in the car, turned the key to the on position and back off but car doesn't stop. turned key back on and engaged starter briefly and back off again, car stopped. took keys out of ignition and put them on seat, stepped out and car started again. at this point i just left it running, connected the jumpers, replaced the battery, reconnected cables to new battery, removed jumpers, put key into ignition and turned fully to engage starter briefly and turned off. engine stopped and has not come back on on its own since. with all this in mind it may have to do with loose battery connections causing the problems but i'm not sure how that could be either, it just seems logically to be more likely given the circumstances - at least in this case.
my wife is out with the car presently or i would loosen things up again and try to replicate, maybe i'll try this when she gets back and post a follow up.

 

Transformers, robots in disguise

 

When battery cable was loosened it could have caused a spark which will zap the electrical system with surge current. In this case precise result is unknow. It can damage something for good or it can damage something partially creating hard to explain funny recurring problems. Today's car electronics are quite sensitive.

 

May be your start button has failed. Easy enough to swap it out and see if it happen again.

 

The cars are going to take over the world!

 

Without brake pedal is pushed down button alone can start the car.