How to make the Twitter API (home_timeline) returns me over 800 tweets? For example applications like tweetbot how do it to have more updates in the 3000 home timeline? I think that using a cache system but which one?
You can not get more then 800 tweets in 1 go.
But you can pass parameters like since_id and max_id: https://dev.twitter.com/rest/reference/get/statuses/home_timeline
If you store the oldest tweet id => and in the next api call you pass in that id, then you will get the older ones. After that you just repeat the cycle.
For example: tweet id 500 of date 20/6/2015 is your oldest entry. => max_id: 500 Will return tweets 499, 498, ... which are all older then 20/6/2015