Requirements:

- Including nested tasks
- Allowing free-text descriptions for tasks without time constraints
- Implementing access levels
- Providing a free subscription for a minimum of 5-10 users with no time limit

Wishes (with added personal input):

- Incorporating a built-in editor for enhanced task management
- Offering seamless integration with Google Docs for efficient collaboration
- Facilitating integration with box.com for file sharing and storage
- Providing both public and private bug/task tracking functionalities
- Offering analytics features for performance evaluation
- Allowing placement on your domain for branding purposes
- Providing hosting options for ease of deployment
- Supporting open-source development for community involvement

Now, I attempted to utilize the free edition of teamlab; however, I found that working with tasks and discussions did not meet my expectations. Additionally, the people module felt lacking in some aspects.

Unfortunately, neither Redmine nor YouTrack are available for installation in the AppFog cloud platform.

Consider trying Asana for your task management needs. While it may lack an editor, it is still highly suitable as a small-scale task manager.

Additionally, we have been using Teambox for approximately a year now, and it has proven to be extremely convenient for collaborative work.

Have you considered Redmine? It appears that there are numerous plugins available for sharing with this platform. Despite some complexity in hosting installation, I have yet to encounter a better tool for work.

After thoroughly researching various tools, birdviewprojects currently stands out as the frontrunner.

If you specifically require "Nested Tasks" and a "Public and Private bug/Task tracker," I recommend giving https://pintask.me/ a try.

I was initially inclined to recommend Basecamp, but I recalled that it is a paid service.

Alternatively, you could consider Wunderlist as an option. However, I have not personally used it for teamwork, although it does offer sharing functionality.

I highly encourage exploring other task management tools that align with your specific requirements and preferences. Efficient task management is crucial for successful collaboration and project completion.

Based on your needs, I suggest considering a customizable project management software such as JIRA.

JIRA allows for the creation of nested tasks and offers free-text descriptions for tasks without time constraints. It also supports access level customization, enabling you to control user permissions. While JIRA does not offer a free subscription for more than 10 users with no time limit, its flexible pricing options can accommodate a small team.

Additionally, JIRA offers a built-in editor for enhanced task management and seamless integration with Google Docs for efficient collaboration. It also supports integration with box.com for file sharing and storage. JIRA's public and private bug/task tracking functionalities, analytics features, and custom branding options can align with your wishes.

To address your desire for open-source development and hosting options, JIRA provides self-hosting capabilities and allows for community involvement through its open-source development model.

 JIRA offers a wide range of features that align with your requirements and wishes.

1. Nested tasks and free-text descriptions: JIRA allows for the creation of sub-tasks, enabling the nesting of tasks within larger projects. Additionally, it supports free-form text for task descriptions, accommodating tasks without specific time constraints.

2. Access levels: JIRA provides robust access control features, allowing you to define user permissions at a granular level. This ensures that sensitive information or critical tasks are accessible only to authorized team members.

3. Free subscription for a small team: While JIRA may not offer a completely free subscription for 5-10 users with no time limit, its flexible pricing options, including a free trial period, can be suitable for small teams.

4. Built-in editor and integration capabilities: JIRA includes a powerful built-in editor for task management, allowing for rich text formatting and attachments. It also seamlessly integrates with Google Docs for efficient collaboration, meeting your wish for enhanced editing capabilities and collaboration tools.

5. Integration with box.com for file sharing and storage: JIRA offers integration with popular file storage and sharing platforms, including box.com, providing a seamless experience for managing project-related files and dоcuments.

6. Bug/task tracking functionalities and analytics: JIRA excels in bug and task tracking, offering both public and private tracking options. Moreover, it provides advanced analytics features, allowing for performance evaluation and project insights.

7. Custom branding and hosting options: JIRA enables custom branding, including the ability to use your own domain, reflecting your organization's identity. Additionally, JIRA supports self-hosting, providing options for deployment according to your hosting preferences.

8. Open-source development and community involvement: JIRA is part of the Atlassian ecosystem, which actively involves the developer community. While not entirely open-source, JIRA allows for customization and extensibility through its developer-friendly architecture.

TeamLab's free edition was a bit of a letdown. The tasks and discussions didn't quite meet expectations, and the people module felt a bit lacking. Redmine and YouTrack are solid options, but the fact that they're not available for installation in the AppFog cloud platform is a major bummer.

Consider Asana, Trello, or Basecamp. These tools offer a free subscription for a minimum of 5-10 users with no time limit, and they're all about flexibility. Asana's got a built-in editor for enhanced task management, while Trello's all about seamless integration with Google Docs. Basecamp's got a sweet spot for file sharing and storage, and it's all about ease of deployment.