I have owned two domestic mini vans previous to the Odyssey, I took a large loss on both when selling them. I thought a Honda would retain more value over its life, so I bought the Odyssey.
It has been a mediocre vehicle as long as I have owned it. It started having transmission problems at 65,000 kilometres, not shifting at all eventually. I had always had all the service work and oil changes done at our Honda dealer so they replaced the transmission on warranty, I paid the bill for the next Transmission at 130,000 kilometres my self ($3500) and it is on the way out again. I see that there was a recall on the transmissions eventually, but I did not qualify as I had too many kilometres, be prepared to replace transmissions every 60,000-70,000 kilometres.
The sliding doors have been a problem since new, if you spray the rubber seals with silicone every couple of weeks it helps to stop them from sticking.
I have replaced the Battery almost every year that I have owned the van, but that only seems to work for a month or two and the battery starts going dead again, the dealer is unable to find a problem with the electrical system.
Overall I have been disapointed with Van, but the good news is the resale value has remained very high.