Most teachers have stated learning objectives for their courses. They describe what we plan to teach and how we plan to assess students.
You may have read this summer about a case involving whether a professor can be required to write those down on a syllabus. A professor at the College of Charleston brought a lawsuit against the school that claimed that he was losing his job for refusing to include learning outcomes (the same as objectives?) in his syllabus.
The answer to the question of whether a course can not have learning objectives is a pretty resounding No. Of course, I'm sure students could point out some truly dreadful courses that did not have clear objectives or outcomes. Whether they are stated explicitly to students, probably in the syllabus, is the real question in that case. My answer to this second question of whether or not a course can not have clearly stated objectives is a resounding Yes.
Faculty need to consciously establish their goals and objectives in designing the course, but they also need to communicate those to students.
I would say that kind of information information should be available to a student before she even signs up for the course, perhaps in a course catalog or online page about the course. The objectives should also be explained in greater detail in the syllabus and in the course itself.
That is an instructional design task. I was very surprised how difficult it was to get faculty that I worked with on course design to understand the difference between a goal and an objective. We can get bogged down and confused in talking about goals, objectives and outcomes. If faculty are confused, certainly the students will be confused as well.
In a bit of an oversimplification, a goal is an overarching principle that guides decision making, while objectives are specific, measurable steps that can be taken to meet the goal.
You can further muddy this academic water by adding similar, but not interchangeable, terms such as competencies and outcomes. In this document I found online and used in some version for faculty workshops, it says: "From an educational standpoint, competencies can be regarded as the logical building blocks upon which assessments of professional development are based. When competencies are identified, a program can effectively determine the learning objectives that should guide the learners’ progress toward their professional goals. Tying these two together will also help identify what needs to be assessed for verification of the program’s quality in its effectiveness towards forming competent learners...In short, objectives say what we want the learners to know and competencies say how we can be certain they know it."
Whatever terminology you use, teachers need to know the larger goals in order to design the ways they will be presented and how they will be measured. Students need to knew as early as possible those last two parts.