In terms of materials, chicken wire mesh will be the most inexpensive option to use when building your fence. The downside is that it’s not the most attractive garden fence option. In a similar vein, you could choose a DIY fence made out of repurposed items, like car tires, or wooden pallets, either of which would be free to collect. 

You can keep a dog from digging under a fence by filling existing holes in your yard, installing a wire to your fence bottom, burying your fence bottom, making a designated digging spot, and removing visual stimuli. You can also dissuade your dog from wanting to dig under your fence and escape by scheduling more one-on-one playtime and ensuring they are getting enough exercise to keep them stimulated.

Wire fences are generally the most inexpensive type of farm fence to install, especially compared to lumber. Barbed wire is one of the cheapest options due to its simple materials, making it a good choice for low budgets and for confining docile animals in large spaces. Other options include hog wire and woven wire fences, which are strong, long-lasting, and relatively easy to repair. While wire fences require less lumber and reduce overall project costs, they are often considered less aesthetically pleasing than traditional wood fences.

Whether you need a permit to build or install a fence depends on several factors, including your location, the fence's height, its placement in your yard, and the materials used. In many cities and highly populated areas, a permit is required, especially for fences taller than 6 feet. Conversely, rural areas may have fewer restrictions. Typically, backyard privacy fences 6 feet or shorter do not require a permit, while front yard fences often have a lower height limit, sometimes as low as 3.5 feet. There can also be exceptions for properties in mixed residential-commercial zones. It's best to check with your local building inspection or zoning department to understand the specific requirements, height restrictions, and location rules for your area. A professional fence installer can help you navigate this process and often secure the necessary permits for you, though this may extend the project timeline.