When you have a successful website that get over 100,000 hits a day, you should indeed begin looking for a dedicated server. Being on a shared solution will cause for the webhost to look at the average system consumption, effect on your neighboring users, and what steps need to be taken to improve the situation (i.e. hardware upgrade) - and if such an upgrade is directly caused by your account. Depending on the answers of these steps, the webhost may impose fees for you (depending on their policies) - as the webhost wants to maintain quick performance to everyone on the server.
While your website is consuming many resources, you may settle for a little slower response time, as you will know that your website is very popular (and so will your visitors) - however, the client on the same server that only get 100 hits per day (and only a quarter are unique), will most likely not accept the slowness of his or her 5 page website, as the end-visitor will not understand why such a small website takes so long to load and simply stop loading and leave.
The webhost, in turn, has the responsibility to make everybody happy (both you and everybody else on the server), and in order to achieve this goal, somebody must cover the extra incurred expenses. The most fair way out, is to indeed bill the consuming client for his or her usage. This is a comparable to the more traditional scenarios of data transfer and disk space - the more you use, the more you pay.
If resources, and nothing else, is the main problem - then you should definitely look for a dedicated server solution.
There are many dedicated servers starting at $99/month, with plenty of RAM and CPU power, that should be looked at.
Of course, you can always give a webhost that specializes in bigger plans a try, and see how it performs and if you will get any notices from that webhost, be it us or somebody else. You can shop around and see who offers what dedicated server that would be within your budget, then signup for a shared solution plan, and if the shared solution plan doesn't work out, simply upgrade to the dedicated server plan - instead of having to change the webhosting provider again.