As with any growing Card Game (LCG, CCG, TCG), unless you want to play competitively, it's never required to get more cards. Since you don't need to keep up with changing metagames, new strategic ideas and new available cards, you can play with just what you have with your friends.
The question remains how much fun you get out of it after having played for a while.
To answer that, I'll point you towards card games with a limited card pool. If you're playing with a fixed set of 32, 52, or just any number of cards, you can easily calculate the maximum number of combinations of cards each player can have. Combining those into tactical clusters (in Poker, for example, it doesn't matter which Aces you have, 2 of them always have a specific strength) reduces that number to something in the 6 figures (more or less). Each of these cluster distributions gives a unique gaming experience and additionally, the fact there is private information involved (you don't know which cards the other player(s) have) further increases the complexity and diversity across several games.
Imagine having more cards. Many more cards.
The more cards you have available, the more complex situations can become. Deckbuilding really only becomes a thing when you have a big pool of cards available, but usually, LCG core sets already offer some number of different options. There are always several decks belonging to different factions you can choose from, sometimes even tweak. What this means is you're not simply dealt a few cards, but you can invest the effort to carefully decide which cards you want to have available in the first place. Then, when you're actually playing, you have a similar situation to the one I described with the fixed deck games, where you have a giant number of possible card distributions, which are reduced by tactical clustering (or several copies of the same cards being played), but diversified by the existance of private information.
So, conclusion time.
In my personal opinion, LCG core sets offer a lot of replayability and long-term fun, but if you really like the game and play a lot, strategy becomes such a strong factor that you quickly learn how each deck works and what its strengths and weaknesses are. At that point, the limited card pool leaves you craving for more options to surprise an equally educated opponent.
My advice is this. Play with the core set. If you really like the game and play a lot, you will reach a point where you get the feeling you've mastered all of that game and it doesn't offer you any more surprise. At that point (or earlier if you just feel like it), get an expansion set. Basically, expansion sets for LCGs are like new, but similar games, except they actually have the size of the core and expansion sets combined.
How long it takes until you should probably get an expansion really depends on how much you play and how long it takes you to figure out a game. If you have any decently complex game which you liked playing but at some point grew sick of, you can take that as a reference. But with an LCG, you can buy an expansion set at that point and revive the fun (and possibly even discover an entirely new aspect of the game in complex deckbuilding).