It most probably wouldn't.  Well, as long as the total market cap remained between $0.000001 and $100, it would simply be a dead project, wouldn't it ?

To have the market cap grow to, say, $10 000, it would mean that people made coins that cost them $10.  They were NOT expecting any significant price rise, so the only way they did it, was that they expected it to remain close to that value.  Most probably, because they needed it as a currency - there's no other reason to spend $10 of work on something that won't be worth much more, right ?

So if the market cap rose to $10 000, it means that people had USE of it as a currency.  They wouldn't have any serious expectations of gain.  Only of use.  Say on dark markets.  On web sites.  On VPN.  On this and on that.  After a while, the price would "stick".  People wouldn't put fortunes in it, they would only acquire what they need to use as a currency.  They wouldn't be watching coinmarketcap to see what goes up and what goes down.  It would just be a real currency.

Of course, there could be moments where there's less demand for currency, that is, when its market shrinks and its adoption falls.  Yes.  I would think that if adoption seriously fell down, and hence, the price with it, there are two possibilities: people have faith in it, and hence, would like to see it as an opportunity to "buy $10 bills for $8", pushing the price back to $10.  Or the thing crumbles down.  Indeed, if during a ramp-up phase, people had only acquired it to buy stuff, got used to its price, and then see a slight dip, it would be "shopping time".  If adoption would, after a slow-down, take off again, price would again rise up to $10, and people would be confirmed in their belief in its price.  Those that "bought the dip" would have been rewarded.  The "stickiness of price" would latch on.  
It wouldn't be a speculative thing.  It would be driven by utility.  As long as the demand for that utility would be steadily increasing, even if sometimes it would drop, the price would be pushed against the $10 limit, even if it dropped sometimes.  That would induce people to hoard it when it drops, because they know there's all the chances in the world it would go up to $10 again if utility demand rises again.