Other contributors have said "How many countries participated in the Athens Olympics?" is the same question as "Which countries competed in the 2004 Olympics?". If you believe that these are not asking the same thing and should be answered differently, click here.
There were 202 countries competing in Athens:
Afghanistan (suspended in 2000) Albania Algeria American Samoa Andorra Angola Antigua and Barbuda Argentina Armenia Aruba Australia Austria Azerbaijan Bahrain Bangladesh Barbados Belarus Belgium Belize Benin Bermuda Bhutan Bolivia Bosnia and Herzegovina Botswana Brazil British Virgin Islands Brunei Bulgaria Burkina Faso Burundi Cambodia Cameroon Canada Cape Verde Cayman Islands Central African Republic Chad Chile China Colombia Comoros Congo, Democratic Republic of the Congo, Republic of the Cook Islands Costa Rica Cote d'Ivoire Croatia Cuba Cyprus Czech Republic Denmark Djibouti Dominica Dominican Republic Ecuador Egypt El Salvador Equatorial Guinea Eritrea Estonia Ethiopia Fiji Finland France Gabon Georgia Germany Ghana Greece Grenada Guam Guatemala Guinea Guinea-Bissau Guyana Haiti Honduras Hong Kong Hungary Iceland India Indonesia International Olympic Athlete (East Timor in 2000) Iran Iraq Ireland Israel Italy Jamaica Japan Jordan Kazakhstan Kenya Kiribati (NOC recognized in 2003) Korea, North (DPR of Korea) Korea, South Kuwait Kyrgyzstan Laos Latvia Lebanon Lesotho Liberia Libya Liechtenstein Lithuania Luxembourg Macedonia, Former Yugoslav Republic of Madagascar Malawi Malaysia Maldives Mali Malta Mauritania Mauritius Mexico Micronesia, Federated States of Moldova Monaco Mongolia Morocco Mozambique Myanmar (Burma) Namibia Nauru Nepal Netherlands Netherlands Antilles New Zealand Nicaragua Niger Nigeria Northern Mariana Islands Norway Oman Pakistan Palau Palestine Panama Papua New Guinea Paraguay Peru Philippines Poland Portugal Puerto Rico Qatar Romania Russia Rwanda Saint Kitts and Nevis Saint Lucia Saint Vincent and the Grenadines Samoa San Marino Sao Tome and Principe Saudi Arabia Senegal Serbia and Montenegro (Yugoslavia) Seychelles Sierra Leone Singapore Slovakia Slovenia Solomon Islands Somalia South Africa Spain Sri Lanka Sudan Suriname Swaziland Sweden Switzerland Syria Taiwan (Chinese Taipei) Tajikistan Tanzania Thailand The Bahamas The Gambia Timor-Leste (participated under IOA in 2000) Togo Tonga Trinidad and Tobago Tunisia Turkey Turkmenistan Uganda Ukraine United Arab Emirates United Kingdom (Great Britain) United States Uruguay Uzbekistan Vanuatu Venezuela Vietnam Virgin Islands Yemen Zambia Zimbabwe
More input:
Puerto Rico is an independent country (free associated state to the USA, but independent) Puerto Rico is not a "territory" of the USA (so far). American Samoa is an USA protectorate (The USA take care after it but it is not part of the USA. Therefore, it is somehow "independent" since it is not a "territory" of the USA.