The reputed universities and colleges always ask for the proof of proficiency in English language if English is not your native language. The English language requirement may be waived if you are educated in English and you are too good in it. But for that you have to request university authorities to grant a waiver showing the documental proof of your language proficiency. If you really intend to study, then my suggestion would be to take IELTS or TOEFL. These tests not only help you to get admission in the well-known universities but also will be beneficial while applying for the visa. These days most of the visa denial cases are because of low TOEFL/IELTS score or no score at all.

There are some universities in USA which offers conditional admission to international students who do not have the required English language proficiency or who have not taken any of such proficiency tests. However, such students will be required to attend English as a Second Language (ESL) classes until they meet the qualification requirements.

The list of some of the universities that offers conditional admission is:

University of Missouri-Kansas City
University of Delaware
University of Scranton
Portland State University
University of Arkansas
University of North Texas
University of Wisconsin, River Falls
University of Oregon
Eastern Washington University
Stony Brook University
Community Colleges of Spokane
Central Michigan University
Oregon State University

You can't actually apply for the Australian study visa without IELTS score.

I don't think any universities in Australia will accept any international students without proficiency in English. IELTS and TOEFL are some ways to know the proficiency in English. Also, if there are any which accepts without scores of TOEFL or IELTS, I bet, they will not be providing good education. So, I think you shouldn't run behind the universities which do not accept you without IELTS or TOEFL. But do either TOEFL or IELTS and hunt for the good universities.