I'll assume living memory to be a nice round number like 60, so we are looking for the closest Native American nation that did not have direct contact with Europeans between 1657 and 1717. I'll also assume that not all members of the group interacted with Europeans, just that somewhat regular contact/influence existed some time between 1657 and 1717.
The closest nations to Boston, the Wampanoag, Nipmuck, Narragansett and Nashaway, were swept up in the hostilities with King Phillip's War from 1675-1678. Any Native American belligerents not killed in the conflict and not professing Christians were sold into slavery in Bermuda. Praying Indians settled in towns like Natick, Grafton and Marlborough, and refugees fled inland to join other nations.
Expanding out of the direct Boston/Providence Plantation area, we have the nations in Maine, New Hampshire, Vermont, and New York. Coastal populations in Maine would have regular trading contact with white merchants and fisherman, so those nations are out. Further inland many nations possibly contacted a European during King William's War (look, a map] during the 1680s and 1690s. I am not as familiar with the small nations of northern Maine, but I assume the constant tension between the French and English colonial enterprises, and the Maine Amerindians strategic importance as allies, meant they were courted extensively during the period. Further north into Canada the Wendat/Huron were strongly allied with the French, and during the diaspora caused by the Beaver Wars (to be explained shortly) fled west to the Great Lakes.
Looking west, here is a generalized map so we can keep our bearings. The Mahican and the Delaware definitely encountered Europeans in the period of interest. First contact along the Mohawk River occurred in 1609 and these groups were intimately involved in the game of colonial expansion with New York/New England/New France.
1657-1717 is a very interesting time for the Haudenosaunee/Iroquois. Hostilities related to the fur trade and disease mortality brought about a series of mourning wars that increased the Haudenosaunee territory, and wrought destruction over a huge swath of the U.S. Midwest. The Beaver Wars turned the Great Lakes into a war zone. Iroquois raids depopulated much of the U.S. Midwest and sent refugees fleeing to the lands bordering the Great Plains. French missionaries fled west with their Huron flock, establishing missions on Lake Michigan by 1652 and the western tip of Lake Superior by 1661. As they moved west, the Huron diaspora opened up the fur trade to nations previously beyond the frontier outposts. Heading directly west of Boston we are basically on the Great Plains before we consider a nation without European contact.
Next, lets dive south. English colonial enterprises in Pennsylvania contacted the Erie, Shawnee and previously mentioned Iroquois. Jamestown was established in 1607, and English influence continued to grow among the Algonquian nations living in the tidewater. To the west of the tidewater we start to run into the area of Iroquois expansion during the Beaver Wars so I will dive further south.
Between 1657 and 1717, traders operating out of Virginia and Carolina colonies united the greater U.S. Southeast based on the trade of Indian slaves and deerskins. Though actual "white guy presence" was limited, the repercussions of the Indian slave trade destabilized the south. Pressure from the French and English spurred the rise of confederacies like the Creek, Cherokee, Chickasaw and Choctaw to deal with the encroachment of the French to the west (operating from Louisiana and the Mississippi River) and the English (with their native, slave-raiding allies) to the east. As an aside, from 1696-1700 a devastating smallpox epidemic spread through the southeast causing tremendous mortality in the region. Epidemics, combined with refugees fleeing the Yamasee War from 1715-1717, shattered the existing lifeways in the south. The interior southeastern nations, as well as the petite nations along the Gulf Coast, all felt the repercussions of contact with Europeans between 1657 and 1717.
So, we are at the Mississippi and we have yet to encounter a Native American nation without contact with/significant influence from Europeans between 1657 and 1717. Obviously, the nations of New Mexico and Texas near Spanish missions are out of consideration. In New Mexico the Spanish presence along the Rio Grande ensures the Pueblos, Apaches, Utes, Dine, Comanche and many smaller Southwestern nations could contact a white guy. The first mission in Spanish Texas, San Francisco de los Tejas, was established in 1690. The Spanish presence in Texas was not influential, long-lived, or particularly productive during our time period, but it does mean we need to look north of the Red River for a nation that fits our criteria (Edit: The first mission founded in the geographic region that we know as Texas was San Angelo, founded in 1632. Thanks to /u/_choupette for clarifying.)
Where does this leave us? In 1717 you would need to be on the Great Plains, likely north of the Red River, and west of French influence along the Mississippi River/Great Lakes to encounter a Native American nation that had yet to contact a white person. Even then, contact was imminent. For example, the first European encounter with the Mandan occurred with the arrival of a French Canadian trader, Sieur de la Verendrye, in 1738.
Sources
Calloway Once Vast Winter Count: The Native American West Before Lewis and Clark
Kelton Epidemics and Enslavement: Biological Catastrophe in the Native Southeast 1492-1715
Gallay The Indian Slave Trade: The Rise of the English Empire in the American South, 1670-1717
Bragdon Native Peoples of Southern New England, 1650-1775
Trigger The Children of Aataentsic: A History of the Huron People to 1660
Weber The Spanish Frontier in North America
This is an interesting question, and I'd like to tack on a follow-up question. How likely would it be to find native peoples living in Boston at the time? Would there be any? If there were, would they be discriminated against?
Alternatively, which tribe or community was the last to be contacted in North America by non-Natives?
This question isn't too hard to answer for me, but the citations are killing me. Knowing about certain native cultures from grad school and books like Red, White, and Black and The American Indian, Past and Present I can rationalize contact points. Combining that with knowledge of Hudson Bay and French Fur Trade operations from my thesis and some background knowledge in Spanish colinzation in North America, I'm pretty confident that the answer is one of the Salish or Sahaptin tribes within the Columbia Plateau in Washington.
The Hudson Bay had been moving through central and southern canada for years interacting with those tribes before crossing the Rocky Mountains. Spain's influence extended up to Oregon and plains tribes were so extremely mobile after 1680 that they would have contacted both Hudson Bay and Spanish people. French Fur trading posts were also extended out to the Dakotas.
It's likely that on a band to band level that you might be able to find tribes in South Dakota or Wyoming with no contact with white people, but on a greater tribe scale it's not likely.
Washington Coastal tribes also had significant contact with Russians and some American Fur Traders but no one was willing to cross the cascades until after Lewis and Clark.
TL;DR Washington State
The short answer is that you would have to cross the Appalachians and head far west, probably across the Mississippi and into the Northern Rockies and Columbia Plateau. Anything closer would have probably had contact with the Spaniards who would have been in and around Santa Fe for over 100 years at that point, or with Acadian voyageurs who were all over the place so long as there was a body of water within a few day's march and who, by 1717 would have had upwards of 50 years head start on you in terms of exploring the interior.
