elastic search, is it possible to update nested objects without updating the entire document?

You can use the Update API.

curl -XPOST localhost:9200/docs/posts/post/_update -d '{
    "script" : "ctx._source.nested_user = updated_nested_user",
    "params" : {
        "updated_nested_user" : {"field": "updated"}
    }
}'

See this SO answer for full detail.

Note that update scripts support conditional logic, as shown here. So you could tag forum posts when the user changes, then iterate over the posts to update only posts with changed users.

curl -XPOST 'localhost:9200/docs/posts/post/_update' -d '{
    "script" : "ctx._source.tags.contains(tag) ? "ctx._source.nested_user = updated_nested_John" : ctx.op = "none"",
    "params" : {
        "tag": "updated_John_tag",
        "updated_nested_John" : {"field": "updated"}
    }
}'

UPDATED

Perhaps my ternary operator example was incomplete. This was not mentioned in the question, but assuming that users change their info in a separate part of the app, it would be nice to apply those changes to the forum posts in one script. Instead of using tags, try checking the user field directly for changes:

curl -XPOST 'localhost:9200/docs/posts/post/_update' -d '{
    "script" : "ctx._source.nested_user.contains(user) ? "ctx._source.nested_user = updated_nested_John" : ctx.op = "none"",
    "params" : {
        "user": "John",
        "updated_nested_John" : {"field": "updated"}
    }
}'

As mentioned, though, this may be a slower operation than reindexing the full posts.

Sadly elasticsearch cannot update only part of a document without reindexing the whole document. So, yes you would need to reindex the whole document to change a nested part.

If you don't have the whole document to hand to resend it, you can just send the part that needs changing using the update API, but be warned there are performance issues.

Answer by @Scott Rice on how to use partial update in this context is very useful, while answer by @ramseykhalaf is more correct in sense that this is not possible without reindexing. If we do partial update, we do reindexing whole document anyway.

However depends on understanding of what "reindexing" is.

If we define reindexing as "resubmitting whole document to ES" - then we can call partial update the solution without reindexing in this sense. If we define reindexing as "recalculating data structures allowing to efficiently search for updated document in index" (which is more correct definition to my understanding), then it always happen.

Please note that whole old copy of document will remain in index after partial update, marked as deleted (until next complete reindex from scratch or "optimize").

To avoid this, child-parent relationship can be used instead of nested objects. Children can be added/deleted/updated without touching the parent document (however this have its cost of course - maintaining child-parent relationship forest in memory etc.).