Aside from widely broadcast one-time pad encrypted/authenticated messages (which provides anonymity for the receiver, not the sender -- that can be bootstrapped), deterministic rotating asymmetric key pairs (say, a non-compromised curve ECC scheme using Key Families) coupled with a relatively-high non-deterministic latency mixnet delivery system using onion/garlic routing for intermix peers goes a long way towards achieving data security, authentication and metadata anonymity for all involved parties. Discovery and routing in such a system is a very complicated problem -- and it's one yet to be solved; though, generally if one is going to such lengths, one has a rendezvous system already in place.
The primary problem is that it's expensive for users -- expensive in terms of time, cost, and complexity; which means usage is low. Consequently, it also makes people targets simply for using the system as the number of those using the system is relatively small -- something one can't really get away from easily unless makes it look like one isn't using the technology. To do that, one then has to go through the effort of creating cover traffic -- and creating consistently good cover traffic (good enough to fool a human analyst, because one is one of thousands, not one of millions/thousands of millions) is immensely difficult and techniques change over time w/ local conditions so it's hard to automate. Life gets really really difficult when survival depends not only on keeping people out but also on keeping them from knowing anything of interest is there in the first place!
Don't forget about stylometry and inadvertent signatures; never communicate in real time, avoid absolutely everything but plain text data if at all possible, write very plainly (say, using only first few thousand most common words in your language) and use stylometry defeating tools (for example, Anonymouth -- though I haven't audited it; others have, but it's still just a thin layer to apply to other work) to prevent others from creating signatures based on the words you use and how you use them. NEVER forget that binary data you send might contain metadata fields to give you away (version numbers, encoding settings -- all seemingly innocuous but possibly unique to you!). Sending images? Make absolutely certain your camera doesn't have sensor glitches that can create a signature. ( see https://www.schneier.com/blog/archives/2006/04/digital_camer... ) Don't forget your surroundings either ( NSFW Language/Topic (4Chan helps track down targets) but incredibly illustrative - https://i.imgur.com/nLCklgZ.jpg ). Sending scans of documents? Most new printers print identifying patterns using steganographic techniques. ( see https://www.eff.org/issues/printers ). Stop and meet someone and both of you brought your mobiles? The fact that your phones traveled together, and where you went, is recorded (CO-TRAVELLER; see https://www.washingtonpost.com/world/national-security/nsa-t... ). The world is INCREDIBLY hostile to anonymity seekers.
Relying on third parties to strip data isn't a workable anonymity solution because you can't trust them to do so, correctly or at all. Not to mention that; but with pervasive internet monitoring (which, thanks to Snowden, we know is real) the mere fact that you've communicated with someone or a site storing your data is stored in a place where it cannot be wiped by any party authorised to participate in your conversation. Generally, if you're not anonymous to the person you're communicating with, until you choose to identify yourself during communications who's contents may then be repudiated (say, an olm/axolotl ratchet) at a later time by publishing that private key, now expired and no longer good for future authentication -- then you won't be anonymous to any party, period.
It's a trust no-one, verify everything type of situation; people don't deal well with that. Pervasive encryption is only the first and easiest step. If you want true (or even just reasonable) anonymity, things get very expensive, very quickly.
