I use SparkPost and built an addon for it which provides full bounced email handling (no additional mailbox required):
Symfony Mailer transport implementation for SparkPost
Pricing details here:
SparkPost provides email delivery services and transactional email for developers at companies of all sizes
pages.sparkpost.com
You get 500 emails per month for free for testing purposes - but I wouldn't try and run a site in production using this - that's not what it's intended for and there could be limits.
Their basic plan is US$20 per month for up to 50,000 emails per month.
Other email providers to look at include: Mailgun, Sendgrid (owned by Twilio), Postmark (owned by ActiveCampaign), Netcore (previously Pepipost), SMTP2Go, MailerSend (owned by MailerLite), Mailtrap, Brevo (previously Sendinblue) ... some might have free plans or cheaper plans that suit your usage requirements.