Is it possible to use CiviCRM when we have a website built with Expression Engine rather than Joomla, Drupal or WordPress?
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You would not be able to have direct interaction between Expression Engine and CiviCRM like Drupal, WordPress or Joomla allows, but you could certainly install CiviCRM on a subdomain of your current website along with one of those three and use it on the subdomain. See https://civicrm.stackexchange.com/a/9385/105 for some great thoughts regarding this.
If you had developer support (your own or hired) you could use CiviCRM's API to have certain information from CiviCRM available on your current website.