if you're talking about "Anonymous" exchanges, i'd recommend to use P2P ones, such as Localbitcoins[1] & Paxful[2]. They offer high-privacy trading solutions, but the tradeoff here is that you'll usually pay very high fees.

If you're willing to submit your ID, and have a bank account/VISA that can do SEPA transfers etc, i'd highly recommend you use Coinbase[3], or Bitstamp[4]. You'll pay almost no fees compared to P2P, but then again, you lose all privacy.

[1] https://localbitcoins.com
[2] https://paxful.com
[3] https://www.coinbase.com
[4] https://www.bitstamp.net

There has been scam accusation thread opened against changenow among others, so, its' not advisable to recommend it to anybody at the moment pending when they will come out and very their selves of the scam level against them. 
https://bitcointalk.org/index.php?topic=5034589.0

Will love to recommend Localbitcoin which personally is my favorite since i joined the cryptocurrency world.

Update.

Got the answer from support: https://i.imgur.com/hgKSbrE.png

So... we have traded in 2017 and now in 2019 he marked trade as fraudlent. But he left the localbitcoins and there's no contact with him and he didn't contact me before. I am stucked, but there's solution: I have to find some 2 years old invoices, which might be hard. I have asked some questions LBC support, waiting their reply, mainly how can I contact him, as they have disabled chat.

I see that's a method to block all accounts on LBC. I mean I can mark payment as fraudlent from any other random trades I have done years ago, then the partner is stucked and sometimes it's not possible to provide few years old documents (depends what was traded). In my case, maybe it would be possible, thanks God.

I'm going to add a few new sections to the comparison, like the following:

-Discord support availability
-Telegram support availability
-2FA (Microsoft/Google/SMS) availability

If any of you have any suggestion(s) regarding these modifications, please lemme know. I'd like to know what I should add and what I shouldn't.

Thanks a lot!

It seems like it'd not be wise to add any such exchange to the list; if I do add that, I'll indirectly be responsible for scamming the people.

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I'm sorry, @Ave_Maximus! I can't add that exchange.

I use this exchange for about six months, I exchange small coins , and there was a situation when the coins did not come to the account , I contacted support and they answered me in 1-2 days , I think this is quite normal since from this exchange large and did not expect

I use Binance and ChangeNOW as an alternative exchange without registration. Binance is the best exchange from this list.

In my mind Binance is the best exchange platform which allows to use wide range of altcoins. Of course from time to time there are some technical problems with this service but in any case it is more related to the immense popularity of Binance.

I also used Binance and Bittrex, but Bittrex blocked my account for several weeks without explaining the reason. I stopped using them and do not trust Binance. You should not keep a lot of money in your account, you can lose it once. I also use ChangeNOW as an exchange without registration, without risking my funds.

Some user claimed they are forced to do KYC verification. Its kind of sucks when you are protecting your privacy and anonymity and then the next day you open Binance now you are required to do some verification or its banned from your countries. After all, that is what centralized exchanges do, it always has some strict regulation because it needs to comply with the law.