We've been looking for a house for you for a bit now.  Your first offer lost out to a cash buyer. Your second offer was at a lower price than another buyer offered.  Your third offer had an inspection contingency while the competing offer did not.  Three offers, three misses.

So now we're walking through a fourth house -- and you are starting to think it might be a bit more exciting than the three houses you tried to purchase.

We check with the listing agent. No offers yet.

This might be a relief... no competition... so perhaps we just offer the asking price with a home inspection contingency?  Maybe that works. But, it might just result in other offers coming in and you losing out on another house.

Here's another strategy to consider...

You've lost three houses. Each one for a different reason -- price, terms, or both. And this fourth house seems like a better fit than all of them. Thus, it's likely that other buyers are thinking (or will be thinking) the same thing and might be moving toward making an offer soon too.

So, maybe the question isn't really "what's the minimum offer I can make?" -- but it should be "what would it take to quickly secure a contract on this house?"

Sometimes offering $5,000 over asking price (for example) -- even when no other offers exist -- combined with waiving the inspection contingency, is enough for a seller to say yes and stop showing the house. 

Now, certainly, you might not be comfortable waiving an inspection -- depending on the house, but the point here is... even if you are the only buyer making an offer (thus far) you might want to make it a strong offer if you want to avoid missing out on buying another house that you really like.