In principle Low Cost Carriers (LCC) like Easyjet, Ryanair, or Vueling offer 1 leg journeys alone. Still they offer some interesting "connecting" opportunities. Especially if you start combining between different LCC's. So far I have refrained from combining LCC tickets, because I guess the penalty when you miss a connecting flight can quite high. Is this perception of mine correct, or are there still ways to buy connecting flights on LCC's?
Low-cost airlines operate on a model called 'point-to-point carrier transit', one implication of which is that every single leg of your journey is a separate one. As Stuart mentions in his answer, some LCCs refer to 'hubs' and in fact many of them (especially Asian ones) do operationally maintain hubs, but that's for their own logistics handling. For customer, most LCCs issue a disclaimer along the following lines in their terms and conditions (following taken from easyJet):
They also given good advice on what you should do in case you do plan to book flights on multiple carriers: leave enough of a gap between them.
If you get travel insurance, check whether it covers missing connecting flights as grounds for compensation claim.
easyJet also offers a premium rate flight fare called 'Flexi' and they offer you a free flight if your easyJet flight is more than 15 minutes late. At higher rate 'Flexi' fares you aren't charged booking fees and are free to make changes to your flight dates within a specified period, but the fares are also higher so in some situations you might just find booking Flex-fare priced tickets the same as buying tickets from a full-cost airline!
If you're planning on transferring between carriers on separate tickets (rather than a single combined ticket issued by a codeshare partner), best make sure there's ample time for the connection to account for delays, baggage handling (as said, it might involve rechecking your bags manually, travel to different terminals, customs, etc. etc.) and things like that. Several hours at least over what you'd want for a codeshare is the minimum.
I don't think you can buy 'connecting' flights as the term connecting in air travel has a specific meaning in that they are two flights that can be booked on a single ticket with luggage transferred. However you can reasonably buy two flights that arrive depart from the same airport.
Things to watch out for:
Changing carriers where you need to change terminals, will require something like this border control -> baggage collection -> customs -> baggage drop off/check in -> security -> border control this happened to a friend of mine when he changed from Fly Dubai to Easy jet in Amman he also got stung with a $20USD for a single entry visa.
Does your travel insurance cover missing a non connecting flight if your not even in the country when it departs?
Missing bags might not get forwarded to your destination as the carrier you took your first flight with may not service your finial destination.
Make sure you have a backup plan so you don't spend your entire trip re-routing yourself around only end up where you started 2 weeks later having seen nothing but the inside of airports and train stations.
How likely are you flights to be delayed? Summer is probably less likely than in winter, are there any events taking place in the area that would make finding temporary accommodation expensive or unavailable such as festivals or sports matches?
From my observations you have two options:
A. Pay more and take a full priced carrier to your destination with the security that goes along with a connecting flight.
B. Spend more time (waiting at airports as a buffer encase flights are delayed collecting / re-checking baggage) and pay less.
Essentially it's a cost benefit analysis, which do you have more to spare, money or time?
I have noticed that Easy jet in their in-flight magazine started referring to certain City's as 'hubs' which suggests to me there could be in the near future true connecting flights with Easy Jet.