There is no fixed number. The most significant factors are going to be threefold:

1. What does your historical link growth look like? (obviously this doesn't apply to a new site)
2. Will Google be able to correlate new link growth with an event? (Social promotion, press release, site re-design, etc.)
3. The quality of the links you are building and the way in which they are built (anchor text, DA, PA, etc).

From what I've seen Google takes a baseline, a "normal rate of growth" for both your site and your niche. If you site has had steady or stagnant link growth for years, and all of a sudden you start building a ton of links, that can be a red flag.

For a brand new site, things are a bit different.

Google is going to try to determine, A. have other sites like yours grown this way, B. are these new links spammy or manipulative, and C. does your link growth correlate with something in the news, some social signal, etc.

Of course, at the same time your site will be analyzed from an on-site perspective as well, so if your on-site SEO isn't up to snuff, or you've got duplicate content or indexing issues, link building may not help much regardless of quantity or velocity.

So again, as EGOL said, there is no fixed number. Just try to see things from Google's point of view and keep your link building above board. If you use quality inbound marketing tactics to build links, you shouldn't see any problems.