Can I push server access logs about an Amazon S3 bucket into the same bucket?

Don't push server access logs about a bucket into the same bucket. If you configured your server access logs this way, then there would be an infinite loop of logs. This is because when you write a log file to a bucket, the bucket is also accessed, which then generates another log. A log file would be generated for every log written to the bucket, which creates a loop. This would create many logs and increase your storage costs.