For analyzing large chunks of data, don't use the API. Use the Stack Exchange Data Explorer (SEDE), that's what it's there for. SEDE data lags real-time by 1 to 5 weeks, but that's no big problem for most statistical purposes.
For intense "Big data" analysis beyond even SEDE's capabilities, use the quarterly data dumps. (Thanks, Tim Stone, for the reminder....
You can get this data easily by scraping or by using the websocket interface that Stack Exchange helpfully leaves available.
If you are making an application or webpage, just fetch stackoverflow.com/10m via your app-language's standard fetch/scrape utilities. EG: cURL if you are using PHP, etc.
For client side applications, make a Greasemonkey/...
Well, the response has been underwhelming so far, but fortunately the powers that be decided to restore the voting tab on the profile page: https://meta.stackoverflow.com/a/130380/182796
So a client-side extension is no longer required.