When my daughter was born 20 months ago, I made an email address for her. My wife and I have been sending emails to her all this time, and over the months, we have sent several emails.
Given that we are with our smartphones all the time, it does not take long to compose and send little notes about things we/she did or said, first experiences, fears, joys, often accompanied with photos or videos. We also make predictions (e.g. left handed or right handed) and send them to her, and intend to send a follow up when things are revealed. Apart from getting to relive parts of her life she cannot remember, we feel she can have a detailed account of our joys as her parents, and perhaps apply it when she is a parent too. Our plan is to give the email login info to her when she leaves home for college or something. It's the long con.
I like the intimacy provided by email, and the freedom of format (as compared to Facebook or Instagram).
I got the idea from Google's commercial "Dear Sophie" back in 2011 or so. I have been wanting to this since!
Some notes based on feedback of posting this in another subreddit:
: Feel free to adapt it to what you see as best for you. It is an idea, not a mandate.
: Please set up a recurring reminder to log in to your child's account periodically so the email provider does not delete the account due to inactivity.
: Overwhelming my daughter with emails will take the fun out of it, even though we never intend to task her with reading all our emails. We also noticed that our frequency of emails has dropped from the initial stages when we were excited as new parents, and is now down to once a month. I am certain it will drop further.