OK you said no 3rd party request/code, but how about a request to Google and your own code?
If you have a Gmail account you can do it with Google Scripts. They host the code for you and it's pretty easy.
Go to https://script.google.com and create a new project, delete the code in the window and paste in the following, replacing the email address with the one you want to get the emails sent to.
function doGet(e){
var from = e.parameter.From;
var to = e.parameter.To;
var body = e.parameter.Body;
var emailAddress = '[email protected]';
var message = body;
var subject = 'SMS from ' + from + ' to ' + to + ' received';
MailApp.sendEmail(emailAddress, subject, message);
var output = ContentService.createTextOutput('<Response/>');
output.setMimeType(ContentService.MimeType.XML);
return output;
}
From the Publish menu choose deploy as web app. Set Execute the app as: to Me, then set Who has access to the app: to Anyone, even anonymous.
Now click deploy and Google will ask you to authenticate your script to allow it to send email as you (but not access your incoming email). Once you have granted permission it will give you the URL for your script.
Paste this URL into your Twilio console as the webhook handler for when a message comes in and set the method to HTTP GET
All this will do is email you the SMS. Change the <Response/>
in the line below to contain TwiML for replying to the sender or whatever else you want Twilio to do when messages come in.
var output = ContentService.createTextOutput('<Response/>');
Obviously this is just a basic example, there's no authentication/security etc so any call to your script URL will send you an email. You should really secure it, although the URL Google generates isn't the kind anyone is likely to stumble upon.