Originally Posted by Aprill
There is really no limit - just keep it natural and a bit diverse and everything should be fine.
Agree with you. Last time I checked, search engines loved it natural. By natural here I mean a jagged low volume graph - a realistic one (5 links today, 2 the following day, none for the next 4 days, 6 after that and so on) - as opposed to a synthetic or straight line graph. If it's a topic that's suddenly trending, then it's reasonable to have thousands of shares, links, tweets, likes, pins and so on springing up in a day. However, if there's no clear reason for the sudden surge in activity then Google is most likely going to interpret this as spam.
There's no fixed number on how many links you can get in a day. If there was, then people are likely to find a way to beat the system and this is exactly what Google doesn't want to happen.