BitCoin was a grand idea. It was a pioneer, but, like all pioneers, it got old and younger contenders outmatched him. He's grandpa Crypto, but let's face it: in a few years we're gonna have to take him out the back and shoot him. Bitcoin has lots of problems, among which:

Very slow transactions (can take days to complete), so it could never serve the general population if it gets adopted;
Huge mining/network maintenance cost, which leads to huge transaction fees;
Inability to handle a large number of transactions per second (less than 5 per second!). The network occasionally reaches 100K+ unconfirmed transactions which take days to complete. There are 21 000+ unconfirmed transactions at the time of writing;
The network is exclusively proof-of-work, which makes it vulnerable to attack;
No marketing plan;
Not new user friendly;
No voting system for development of the coin, which decreases user trust and could lead to hard forks.

Because of all these issues, I think it's safe to assume that BitCoin will not be the main cryptocurrency of the future. For it to remain number one, a miracle would have to happen in its development.

That being said, let's jump into what to look for in a currency that has the potential to serve a much larger user base than BitCoin. Obviously, most of them are just solutions to Bitcoin's problems:

Very fast transactions and confirmations;
Acceptable network maintenance costs and much smaller transaction fees;
Can handle a huge number of transactions per second.
A network that is at least partially based on proof-of-stake or importance, so it would be incredibly expensive to attack the network. It would also allow regular users to vote on the future of the network, not just huge players.
A strong marketing plan to attract more users. This should be funded by the network itself through a percentage of the transaction fees going to marketing, not left to charity.
Ease of use for new users. This includes the existence of light wallets, clear explanation of the wallet functions and as little technical details as possible on the front page, so as not to overwhelm the average Joe;
A clearly defined community voting system for major network decisions.
Non-anonymous, experienced developers;
Ease of buying. If you can't buy it with cash or BitCoin without registering an account at an online exchange, then it's not gonna get bought by the aforementioned average Joe;
Transparency. The default transaction type for the currency should be public, otherwise it will look too shady for governments, and for a lot of users too;
It should be designed as a currency, not as a service. Therefor, Ethereum, Augur, Golem etc. don't qualify;
A good name. I can't imagine PotCoin, Pepe Cash or Hacker Gold ever becoming a mainstream currency of the world, can you?

Now that we covered all of that, here are the coins that stand the best chance to overtake BitCoin in the long run, in my personal opinion:

#1 & #2, in no particular order: Dash and NEM

They both have great technology and really quick transaction times, but NEM is a little bit better overall, especially when it comes to the costs of maintaining the network, and therefor transaction fees. However, Dash has better marketing and a higher market cap, and, as we've seen in BitCoin, a large market cap can take an inferior technology a long way. Dash also has optional private transactions, which NEM does not. In my eyes, it's a really close battle between these two and I don't have enough information at the moment to try and predict a winner.

#3: PIVX

As a technology, PIVX is an improvement over DASH in every possible way, but it has a few problems that set it behind its competitors:

No light wallet on the official webpage of PIVX, because none exists yet, although the currency is over a year old. Average Joe would most definitely not like that.
No way of buying the currency without registering at a crypto exchange. While both Dash and NEM can be bought with BitCoin and other cryptos on ShapeShift.io (Dash) or changelly.com (NEM), which are instant exchanges with no account needed, PIVX does not havs that option. This means that you have to pay a transaction fee 3 times when exchanging: when you deposit, when you exchange and when you withdraw, plus the inconvenience of registration. If I wanted to pay huge fees and register to make a transaction, I'd use PayPal instead.
The 2 previous points lead me to my next point, which is: the developers don't seem to be as dedicated as the ones of Dash and NEM, and their marketing skills seem to be inferior. Of course, this is just one man's opinion, so take it with a grain of salt.

I think that, because of these reasons, it would be hard for PIVX to catch up with the competition, unless the developers really start putting some work into it or hire some help.

It is my opinion that one of these 3 currencies will overtake BitCoin in the next 5 to 10 years, multiplying in value hundreds, if not thousands of times.

Even if you don't agree 100% with my picks, I hope this article still provided some useful information to you. Don't forget to follow me if you would like to see my next articles, and upvote my post if you found it interesting. Good luck!