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There are a few good wallets that don't ever hold your keys (so you better!) that also allow many many currencies. Someone will correct me if I'm wrong, but I've found the most adaptable wallets to be Jaxx for smaller coins and the big big ones, and Coinomi will hold almost everything else that most people buy or trade.

Some coins still don't have enough support or the right setup to be in some wallets so they have their own. I would stay away from freewallet.org stuff though....they don't necessarily give you full control, and they are very slow.

JAXX and COINOMI are worth looking into. I've been using Jaxx for a while now, and they keep adding more support for more coins. Coinomi I don't use much but keep it around for moving things around sometimes. Jaxx can hold Tokens as well BTW....well all Ethereum accounts can unless stated otherwise, you just can't always access them unless either supported by the wallet or on an Ethereum web wallet access site....or MEW.

The main thing is that you have the only access to your keys, and that you know your coins are in your wallet, and not on some server somewhere and just represented in your wallet.

Do you send your coins to these wallets after every purchase on the exchange? I don’t have a wallet currently so I guess my coins just sit on exchanges which I’m starting to think is a bad idea. That being said these exchanges hit you with huge fees every time you send coins places. Is storing on Openledge similar to storing in a wallet?

No, storing on OpenLedger is not the same as storing in a wallet, it is the same as storing on an exchange.

I am an active trader, so I do not move funds into and out of wallets often. I have coins I hold long term, and they may be in a wallet, but ones I am trading remain on the platform I am trading on unless I have another need for them. Keeping your money anywhere but your own pocket is always a risk. Most reputable exchanges do not charge anything but the standard fee to move money in and out. You will find high fees anywhere you can use fiat to purchase coins though.

"Do you send your coins to these wallets after every purchase on the exchange?" - Yes, unless you are planning to sell in the very near future.

They have a great many listed and an option to add coins. So I don't know if they can actually take all coins. You can use JAXX as well, but also not all coins are accepted.

MEW only takes ethereum tokens. Many coins are ethereum tokens, which are accepted by MEW.