Usually, there are 2 cases:
1. The 250/10,000' is the general ICAO/FAA restriction, not specific to the SID or Airport. Then normally you could fly the minimum clean, just let ATC know. "xxx departure, our minimum clean today is 265". You can also ask for "high speed climb" or "high speed below 10" and few were the times I was refused.
Note: In the US you can't ask and you won't get high speed climb, only minimum clean.
2. The speed limitation is specific for the airport or SID. Then there is a specific reason and I would not want to go faster than published. If I really need to I will ask ATC but not exceed it without permission.