Cornbread,
The answer to your question will vary with the circumstances. When receiving your initial clearance, you will generally be either cleared to your destination, or in certain circumstances, to a specific local fix. In the case of a local fix, this constitutes your clearance limit, and you will be further cleared in flight. However, when given an initial clearance to your destination, the destination represents the clearance limit until ammended.
See FAA Order 7110-65M, Section 8, Paragraph 4-8-2.
(4-8-2 Clearance Limit: Issue approach or other clearances, as required, specifying the destination airport as the clearance limit if airport traffic control service is not provided even though this is a repetition of the initial clearance.).
When cleared for an approach, the destination for which the approach is designated, is your clearance limit. (AIM 5-2-3(c)). In the event you are unable to complete this approach and must execute the missed approach proceedure, the full missed approach proceedure constitutes the clearance limit. The exception to this is a practice instrument approach, in which the missed approach proceedure is specifically excluded unless otherwise authorized, as specified in AIM 4-3-21(e).
Upon execution of the missed approach proceedure, the pilot bears the responsibility of determining the next course of action, and advising ATC of his or her intentions. See AIM 5-4-19(e). It's best to make this request when announcing the missed approach at or just after the MAP.
Your question specifically asks about the clearance "Cleared Approach," or as you stated, "Cleared for the approach." The latter phraseology does not represent a valid clearance, and would require identification of a specific approach in which the full missed approach proceedure is appended. However, when simply "Cleared Approach," the flight is directed to fly any of the available approaches for that airport. In such a case, the airport remains the clearance limit until a missed approach is initiated, and the ending fix of the specific missed approach proceedure flown will constitute the clearance limit.
You asked about a visual approach. A visual approach does not include a missed approach proceedure, and accordingly, the destination field where the visual approach is conducted constitutes the clearance limit, unless otherwise specified in the clearance. See AIM 5-4-20(e) for clarification.
Bear in mind that a specific clearance my identify other potential clearance limits, ammended missed approach proceedures, alternate instructions, etc.